8 12

8_11

8_13

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 12 at Knotilus!

In symmetric decorative form

Knot presentations

Planar diagram presentation	X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13
Gauss code	1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6
Dowker-Thistlethwaite code	4 8 14 10 2 16 6 12
Conway Notation	[2222]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 8, width is 5, Braid index is 5		[{3, 9}, {4, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 5}, {10, 6}, {9, 4}, {5, 10}, {8, 1}]

[edit Notes on presentations of 8 12]

Knot 8_12.

A graph, knot 8_12.

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["8 12"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6

In[6]:=

DTCode[K]

Out[6]=

4 8 14 10 2 16 6 12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[2222]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(5,\{-1,2,-1,-3,2,4,-3,4\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 5, 8, 5 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 9}, {4, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 5}, {10, 6}, {9, 4}, {5, 10}, {8, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-5] Hyperbolic Volume 8.93586 A-Polynomial See Data:8 12/A-polynomial

[edit Notes for 8 12's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant 0

[edit Notes for 8 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-7 t+13-7 t^{-1} + t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4-3 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 29, 0 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }
HOMFLY-PT polynomial (db, data sources)	$a^{4}-2z^{2}a^{2}-a^{2}+z^{4}+z^{2}+1-2z^{2}a^{-2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 a z^5+2 z^5 a^{-1} +2 z^5 a^{-3} +a^4 z^4-a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -4 z^4-3 a^3 z^3-3 a z^3-3 z^3 a^{-1} -3 z^3 a^{-3} -2 a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +a^3 z+z a^{-3} +a^4+a^2+ a^{-2} + a^{-4} +1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}+q^{12}-q^{10}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-q^{64}+3 q^{62}-3 q^{60}+2 q^{58}-3 q^{54}+9 q^{52}-11 q^{50}+12 q^{48}-8 q^{46}+10 q^{42}-17 q^{40}+23 q^{38}-18 q^{36}+8 q^{34}+4 q^{32}-16 q^{30}+17 q^{28}-13 q^{26}+3 q^{24}+6 q^{22}-12 q^{20}+9 q^{18}-12 q^{14}+21 q^{12}-23 q^{10}+15 q^8-q^6-14 q^4+27 q^2-29+27 q^{-2} -14 q^{-4} - q^{-6} +15 q^{-8} -23 q^{-10} +21 q^{-12} -12 q^{-14} +9 q^{-18} -12 q^{-20} +6 q^{-22} +3 q^{-24} -13 q^{-26} +17 q^{-28} -16 q^{-30} +4 q^{-32} +8 q^{-34} -18 q^{-36} +23 q^{-38} -17 q^{-40} +10 q^{-42} -8 q^{-46} +12 q^{-48} -11 q^{-50} +9 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }

Further Quantum Invariants

Further quantum knot invariants for 8_12.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9-q^7+2 q^5-q^3- q^{-3} +2 q^{-5} - q^{-7} + q^{-9} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}-q^{24}-q^{22}+4 q^{20}-2 q^{18}-5 q^{16}+7 q^{14}-7 q^{10}+5 q^8+2 q^6-4 q^4+q^2+3+ q^{-2} -4 q^{-4} +2 q^{-6} +5 q^{-8} -7 q^{-10} +7 q^{-14} -5 q^{-16} -2 q^{-18} +4 q^{-20} - q^{-22} - q^{-24} + q^{-26} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{51}-q^{49}-q^{47}+q^{45}+3 q^{43}-2 q^{41}-7 q^{39}+2 q^{37}+12 q^{35}+q^{33}-16 q^{31}-7 q^{29}+20 q^{27}+12 q^{25}-20 q^{23}-17 q^{21}+16 q^{19}+22 q^{17}-12 q^{15}-20 q^{13}+7 q^{11}+18 q^9-2 q^7-13 q^5-4 q^3+9 q+9 q^{-1} -4 q^{-3} -13 q^{-5} -2 q^{-7} +18 q^{-9} +7 q^{-11} -20 q^{-13} -12 q^{-15} +22 q^{-17} +16 q^{-19} -17 q^{-21} -20 q^{-23} +12 q^{-25} +20 q^{-27} -7 q^{-29} -16 q^{-31} + q^{-33} +12 q^{-35} +2 q^{-37} -7 q^{-39} -2 q^{-41} +3 q^{-43} + q^{-45} - q^{-47} - q^{-49} + q^{-51} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-q^{82}-q^{80}+q^{78}+3 q^{74}-4 q^{72}-5 q^{70}+3 q^{68}+5 q^{66}+14 q^{64}-9 q^{62}-22 q^{60}-7 q^{58}+11 q^{56}+44 q^{54}+4 q^{52}-40 q^{50}-42 q^{48}-7 q^{46}+77 q^{44}+44 q^{42}-32 q^{40}-76 q^{38}-49 q^{36}+76 q^{34}+80 q^{32}+5 q^{30}-78 q^{28}-82 q^{26}+44 q^{24}+81 q^{22}+33 q^{20}-50 q^{18}-75 q^{16}+8 q^{14}+52 q^{12}+42 q^{10}-15 q^8-49 q^6-20 q^4+18 q^2+41+18 q^{-2} -20 q^{-4} -49 q^{-6} -15 q^{-8} +42 q^{-10} +52 q^{-12} +8 q^{-14} -75 q^{-16} -50 q^{-18} +33 q^{-20} +81 q^{-22} +44 q^{-24} -82 q^{-26} -78 q^{-28} +5 q^{-30} +80 q^{-32} +76 q^{-34} -49 q^{-36} -76 q^{-38} -32 q^{-40} +44 q^{-42} +77 q^{-44} -7 q^{-46} -42 q^{-48} -40 q^{-50} +4 q^{-52} +44 q^{-54} +11 q^{-56} -7 q^{-58} -22 q^{-60} -9 q^{-62} +14 q^{-64} +5 q^{-66} +3 q^{-68} -5 q^{-70} -4 q^{-72} +3 q^{-74} + q^{-78} - q^{-80} - q^{-82} + q^{-84} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{125}-q^{123}-q^{121}+q^{119}+q^{113}-2 q^{111}-4 q^{109}+3 q^{107}+7 q^{105}+5 q^{103}-13 q^{99}-19 q^{97}-6 q^{95}+23 q^{93}+39 q^{91}+22 q^{89}-23 q^{87}-67 q^{85}-59 q^{83}+8 q^{81}+95 q^{79}+114 q^{77}+28 q^{75}-105 q^{73}-174 q^{71}-96 q^{69}+88 q^{67}+233 q^{65}+180 q^{63}-46 q^{61}-258 q^{59}-266 q^{57}-31 q^{55}+253 q^{53}+337 q^{51}+116 q^{49}-221 q^{47}-367 q^{45}-192 q^{43}+157 q^{41}+366 q^{39}+250 q^{37}-93 q^{35}-333 q^{33}-262 q^{31}+25 q^{29}+273 q^{27}+260 q^{25}+20 q^{23}-207 q^{21}-230 q^{19}-56 q^{17}+140 q^{15}+196 q^{13}+81 q^{11}-80 q^9-157 q^7-104 q^5+27 q^3+127 q+127 q^{-1} +27 q^{-3} -104 q^{-5} -157 q^{-7} -80 q^{-9} +81 q^{-11} +196 q^{-13} +140 q^{-15} -56 q^{-17} -230 q^{-19} -207 q^{-21} +20 q^{-23} +260 q^{-25} +273 q^{-27} +25 q^{-29} -262 q^{-31} -333 q^{-33} -93 q^{-35} +250 q^{-37} +366 q^{-39} +157 q^{-41} -192 q^{-43} -367 q^{-45} -221 q^{-47} +116 q^{-49} +337 q^{-51} +253 q^{-53} -31 q^{-55} -266 q^{-57} -258 q^{-59} -46 q^{-61} +180 q^{-63} +233 q^{-65} +88 q^{-67} -96 q^{-69} -174 q^{-71} -105 q^{-73} +28 q^{-75} +114 q^{-77} +95 q^{-79} +8 q^{-81} -59 q^{-83} -67 q^{-85} -23 q^{-87} +22 q^{-89} +39 q^{-91} +23 q^{-93} -6 q^{-95} -19 q^{-97} -13 q^{-99} +5 q^{-103} +7 q^{-105} +3 q^{-107} -4 q^{-109} -2 q^{-111} + q^{-113} + q^{-119} - q^{-121} - q^{-123} + q^{-125} }

A2 Invariants.

Weight	Invariant
1,0	$q^{14}+q^{12}-q^{10}+q^{8}-q^{4}+q^{2}-1+q^{-2}-q^{-4}+q^{-8}-q^{-10}+q^{-12}+q^{-14}$
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-2 q^{34}+6 q^{32}-10 q^{30}+19 q^{28}-30 q^{26}+42 q^{24}-54 q^{22}+64 q^{20}-70 q^{18}+62 q^{16}-46 q^{14}+23 q^{12}+10 q^{10}-44 q^8+80 q^6-106 q^4+124 q^2-130+124 q^{-2} -106 q^{-4} +80 q^{-6} -44 q^{-8} +10 q^{-10} +23 q^{-12} -46 q^{-14} +62 q^{-16} -70 q^{-18} +64 q^{-20} -54 q^{-22} +42 q^{-24} -30 q^{-26} +19 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}+q^{34}-2 q^{30}+3 q^{26}-4 q^{22}-q^{20}+4 q^{18}+2 q^{16}-5 q^{14}+4 q^{10}-q^8-2 q^6+q^4+2 q^2+2 q^{-2} + q^{-4} -2 q^{-6} - q^{-8} +4 q^{-10} -5 q^{-14} +2 q^{-16} +4 q^{-18} - q^{-20} -4 q^{-22} +3 q^{-26} -2 q^{-30} + q^{-34} + q^{-36} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-q^{26}+q^{24}+3 q^{22}-3 q^{20}+q^{18}+5 q^{16}-6 q^{14}+4 q^{10}-5 q^8-q^6+3 q^4+q^2+ q^{-2} +3 q^{-4} - q^{-6} -5 q^{-8} +4 q^{-10} -6 q^{-14} +5 q^{-16} + q^{-18} -3 q^{-20} +3 q^{-22} + q^{-24} - q^{-26} + q^{-28} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{19}+q^{17}+q^{15}-q^{13}+q^{11}-q^9-q^5+q^3+ q^{-3} - q^{-5} - q^{-9} + q^{-11} - q^{-13} + q^{-15} + q^{-17} + q^{-19} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-q^{26}+3 q^{24}-3 q^{22}+5 q^{20}-5 q^{18}+5 q^{16}-4 q^{14}+2 q^{12}-3 q^8+5 q^6-7 q^4+9 q^2-10+9 q^{-2} -7 q^{-4} +5 q^{-6} -3 q^{-8} +2 q^{-12} -4 q^{-14} +5 q^{-16} -5 q^{-18} +5 q^{-20} -3 q^{-22} +3 q^{-24} - q^{-26} + q^{-28} }
1,0	$q^{46}-q^{42}-q^{40}+2q^{38}+3q^{36}-4q^{32}-2q^{30}+4q^{28}+5q^{26}-2q^{24}-6q^{22}-q^{20}+5q^{18}+2q^{16}-4q^{14}-3q^{12}+2q^{10}+3q^{8}-q^{6}-3q^{4}+q^{2}+5+q^{-2}-3q^{-4}-q^{-6}+3q^{-8}+2q^{-10}-3q^{-12}-4q^{-14}+2q^{-16}+5q^{-18}-q^{-20}-6q^{-22}-2q^{-24}+5q^{-26}+4q^{-28}-2q^{-30}-4q^{-32}+3q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-q^{64}+3q^{62}-3q^{60}+2q^{58}-3q^{54}+9q^{52}-11q^{50}+12q^{48}-8q^{46}+10q^{42}-17q^{40}+23q^{38}-18q^{36}+8q^{34}+4q^{32}-16q^{30}+17q^{28}-13q^{26}+3q^{24}+6q^{22}-12q^{20}+9q^{18}-12q^{14}+21q^{12}-23q^{10}+15q^{8}-q^{6}-14q^{4}+27q^{2}-29+27q^{-2}-14q^{-4}-q^{-6}+15q^{-8}-23q^{-10}+21q^{-12}-12q^{-14}+9q^{-18}-12q^{-20}+6q^{-22}+3q^{-24}-13q^{-26}+17q^{-28}-16q^{-30}+4q^{-32}+8q^{-34}-18q^{-36}+23q^{-38}-17q^{-40}+10q^{-42}-8q^{-46}+12q^{-48}-11q^{-50}+9q^{-52}-3q^{-54}+2q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 12"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-7 t+13-7 t^{-1} + t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4-3 z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 29, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{4}-2z^{2}a^{2}-a^{2}+z^{4}+z^{2}+1-2z^{2}a^{-2}-a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 a z^5+2 z^5 a^{-1} +2 z^5 a^{-3} +a^4 z^4-a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -4 z^4-3 a^3 z^3-3 a z^3-3 z^3 a^{-1} -3 z^3 a^{-3} -2 a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +a^3 z+z a^{-3} +a^4+a^2+ a^{-2} + a^{-4} +1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 12"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $t^{2}-7t+13-7t^{-1}+t^{-2}$, $q^{4}-2q^{3}+4q^{2}-5q+5-5q^{-1}+4q^{-2}-2q^{-3}+q^{-4}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-3, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-12$ $0$ $72$ $82$ $30$ $0$ $0$ $0$ $0$ $-288$ $0$ $-984$ $-360$ $-{\frac {8031}{10}}$ ${\frac {1226}{15}}$ $-{\frac {9262}{15}}$ ${\frac {479}{6}}$ $-{\frac {1311}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 8 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 9                 1 1 7               1   -1 5             3 1   2 3           2 1     -1 1         3 3       0 -1       3 3         0 -3     1 2           -1 -5   1 3             2 -7   1               -1 -9 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-2 q^{11}+6 q^9-8 q^8-3 q^7+18 q^6-15 q^5-10 q^4+30 q^3-18 q^2-16 q+35-16 q^{-1} -18 q^{-2} +30 q^{-3} -10 q^{-4} -15 q^{-5} +18 q^{-6} -3 q^{-7} -8 q^{-8} +6 q^{-9} -2 q^{-11} + q^{-12} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-2 q^{23}+2 q^{21}+3 q^{20}-7 q^{19}-5 q^{18}+11 q^{17}+13 q^{16}-18 q^{15}-22 q^{14}+20 q^{13}+40 q^{12}-26 q^{11}-54 q^{10}+23 q^9+73 q^8-20 q^7-88 q^6+15 q^5+100 q^4-9 q^3-108 q^2+4 q+109+4 q^{-1} -108 q^{-2} -9 q^{-3} +100 q^{-4} +15 q^{-5} -88 q^{-6} -20 q^{-7} +73 q^{-8} +23 q^{-9} -54 q^{-10} -26 q^{-11} +40 q^{-12} +20 q^{-13} -22 q^{-14} -18 q^{-15} +13 q^{-16} +11 q^{-17} -5 q^{-18} -7 q^{-19} +3 q^{-20} +2 q^{-21} -2 q^{-23} + q^{-24} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}-2 q^{39}+2 q^{37}-q^{36}+4 q^{35}-9 q^{34}-q^{33}+10 q^{32}+q^{31}+13 q^{30}-32 q^{29}-14 q^{28}+25 q^{27}+19 q^{26}+46 q^{25}-72 q^{24}-58 q^{23}+23 q^{22}+54 q^{21}+130 q^{20}-105 q^{19}-134 q^{18}-21 q^{17}+81 q^{16}+255 q^{15}-101 q^{14}-209 q^{13}-104 q^{12}+77 q^{11}+381 q^{10}-64 q^9-257 q^8-187 q^7+52 q^6+464 q^5-20 q^4-267 q^3-244 q^2+18 q+493+18 q^{-1} -244 q^{-2} -267 q^{-3} -20 q^{-4} +464 q^{-5} +52 q^{-6} -187 q^{-7} -257 q^{-8} -64 q^{-9} +381 q^{-10} +77 q^{-11} -104 q^{-12} -209 q^{-13} -101 q^{-14} +255 q^{-15} +81 q^{-16} -21 q^{-17} -134 q^{-18} -105 q^{-19} +130 q^{-20} +54 q^{-21} +23 q^{-22} -58 q^{-23} -72 q^{-24} +46 q^{-25} +19 q^{-26} +25 q^{-27} -14 q^{-28} -32 q^{-29} +13 q^{-30} + q^{-31} +10 q^{-32} - q^{-33} -9 q^{-34} +4 q^{-35} - q^{-36} +2 q^{-37} -2 q^{-39} + q^{-40} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-2 q^{59}+2 q^{57}-q^{56}+2 q^{54}-5 q^{53}-2 q^{52}+9 q^{51}+3 q^{50}-2 q^{49}-3 q^{48}-18 q^{47}-8 q^{46}+22 q^{45}+32 q^{44}+14 q^{43}-20 q^{42}-63 q^{41}-52 q^{40}+30 q^{39}+99 q^{38}+101 q^{37}-q^{36}-149 q^{35}-185 q^{34}-39 q^{33}+177 q^{32}+285 q^{31}+144 q^{30}-202 q^{29}-411 q^{28}-251 q^{27}+169 q^{26}+520 q^{25}+428 q^{24}-118 q^{23}-632 q^{22}-588 q^{21}+23 q^{20}+695 q^{19}+777 q^{18}+91 q^{17}-748 q^{16}-931 q^{15}-217 q^{14}+766 q^{13}+1064 q^{12}+339 q^{11}-761 q^{10}-1171 q^9-444 q^8+743 q^7+1238 q^6+535 q^5-705 q^4-1286 q^3-605 q^2+666 q+1291+666 q^{-1} -605 q^{-2} -1286 q^{-3} -705 q^{-4} +535 q^{-5} +1238 q^{-6} +743 q^{-7} -444 q^{-8} -1171 q^{-9} -761 q^{-10} +339 q^{-11} +1064 q^{-12} +766 q^{-13} -217 q^{-14} -931 q^{-15} -748 q^{-16} +91 q^{-17} +777 q^{-18} +695 q^{-19} +23 q^{-20} -588 q^{-21} -632 q^{-22} -118 q^{-23} +428 q^{-24} +520 q^{-25} +169 q^{-26} -251 q^{-27} -411 q^{-28} -202 q^{-29} +144 q^{-30} +285 q^{-31} +177 q^{-32} -39 q^{-33} -185 q^{-34} -149 q^{-35} - q^{-36} +101 q^{-37} +99 q^{-38} +30 q^{-39} -52 q^{-40} -63 q^{-41} -20 q^{-42} +14 q^{-43} +32 q^{-44} +22 q^{-45} -8 q^{-46} -18 q^{-47} -3 q^{-48} -2 q^{-49} +3 q^{-50} +9 q^{-51} -2 q^{-52} -5 q^{-53} +2 q^{-54} - q^{-56} +2 q^{-57} -2 q^{-59} + q^{-60} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-2 q^{83}+2 q^{81}-q^{80}-2 q^{78}+6 q^{77}-6 q^{76}-3 q^{75}+11 q^{74}-q^{73}-2 q^{72}-12 q^{71}+11 q^{70}-16 q^{69}-7 q^{68}+38 q^{67}+16 q^{66}+3 q^{65}-41 q^{64}+q^{63}-68 q^{62}-33 q^{61}+98 q^{60}+93 q^{59}+77 q^{58}-58 q^{57}-37 q^{56}-237 q^{55}-183 q^{54}+124 q^{53}+255 q^{52}+331 q^{51}+93 q^{50}+5 q^{49}-556 q^{48}-604 q^{47}-102 q^{46}+356 q^{45}+782 q^{44}+595 q^{43}+402 q^{42}-827 q^{41}-1294 q^{40}-786 q^{39}+85 q^{38}+1183 q^{37}+1419 q^{36}+1336 q^{35}-712 q^{34}-1957 q^{33}-1851 q^{32}-715 q^{31}+1194 q^{30}+2244 q^{29}+2655 q^{28}-93 q^{27}-2262 q^{26}-2933 q^{25}-1845 q^{24}+746 q^{23}+2752 q^{22}+3943 q^{21}+806 q^{20}-2153 q^{19}-3708 q^{18}-2911 q^{17}+70 q^{16}+2891 q^{15}+4876 q^{14}+1634 q^{13}-1812 q^{12}-4100 q^{11}-3656 q^{10}-556 q^9+2791 q^8+5381 q^7+2212 q^6-1428 q^5-4191 q^4-4055 q^3-1036 q^2+2568 q+5533+2568 q^{-1} -1036 q^{-2} -4055 q^{-3} -4191 q^{-4} -1428 q^{-5} +2212 q^{-6} +5381 q^{-7} +2791 q^{-8} -556 q^{-9} -3656 q^{-10} -4100 q^{-11} -1812 q^{-12} +1634 q^{-13} +4876 q^{-14} +2891 q^{-15} +70 q^{-16} -2911 q^{-17} -3708 q^{-18} -2153 q^{-19} +806 q^{-20} +3943 q^{-21} +2752 q^{-22} +746 q^{-23} -1845 q^{-24} -2933 q^{-25} -2262 q^{-26} -93 q^{-27} +2655 q^{-28} +2244 q^{-29} +1194 q^{-30} -715 q^{-31} -1851 q^{-32} -1957 q^{-33} -712 q^{-34} +1336 q^{-35} +1419 q^{-36} +1183 q^{-37} +85 q^{-38} -786 q^{-39} -1294 q^{-40} -827 q^{-41} +402 q^{-42} +595 q^{-43} +782 q^{-44} +356 q^{-45} -102 q^{-46} -604 q^{-47} -556 q^{-48} +5 q^{-49} +93 q^{-50} +331 q^{-51} +255 q^{-52} +124 q^{-53} -183 q^{-54} -237 q^{-55} -37 q^{-56} -58 q^{-57} +77 q^{-58} +93 q^{-59} +98 q^{-60} -33 q^{-61} -68 q^{-62} + q^{-63} -41 q^{-64} +3 q^{-65} +16 q^{-66} +38 q^{-67} -7 q^{-68} -16 q^{-69} +11 q^{-70} -12 q^{-71} -2 q^{-72} - q^{-73} +11 q^{-74} -3 q^{-75} -6 q^{-76} +6 q^{-77} -2 q^{-78} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{112}-2 q^{111}+2 q^{109}-q^{108}-2 q^{106}+2 q^{105}+5 q^{104}-7 q^{103}-q^{102}+7 q^{101}-q^{100}-12 q^{98}-2 q^{97}+17 q^{96}-13 q^{95}+3 q^{94}+22 q^{93}+7 q^{92}+7 q^{91}-43 q^{90}-35 q^{89}+10 q^{88}-26 q^{87}+25 q^{86}+81 q^{85}+63 q^{84}+69 q^{83}-77 q^{82}-148 q^{81}-104 q^{80}-156 q^{79}+17 q^{78}+206 q^{77}+282 q^{76}+356 q^{75}+55 q^{74}-268 q^{73}-435 q^{72}-661 q^{71}-340 q^{70}+204 q^{69}+654 q^{68}+1130 q^{67}+788 q^{66}+49 q^{65}-750 q^{64}-1693 q^{63}-1545 q^{62}-598 q^{61}+680 q^{60}+2252 q^{59}+2499 q^{58}+1547 q^{57}-198 q^{56}-2717 q^{55}-3672 q^{54}-2860 q^{53}-655 q^{52}+2844 q^{51}+4768 q^{50}+4537 q^{49}+2104 q^{48}-2562 q^{47}-5828 q^{46}-6373 q^{45}-3891 q^{44}+1784 q^{43}+6441 q^{42}+8212 q^{41}+6132 q^{40}-511 q^{39}-6764 q^{38}-9928 q^{37}-8380 q^{36}-1108 q^{35}+6552 q^{34}+11294 q^{33}+10685 q^{32}+2988 q^{31}-6052 q^{30}-12328 q^{29}-12727 q^{28}-4870 q^{27}+5238 q^{26}+12969 q^{25}+14476 q^{24}+6663 q^{23}-4286 q^{22}-13308 q^{21}-15867 q^{20}-8241 q^{19}+3344 q^{18}+13388 q^{17}+16876 q^{16}+9543 q^{15}-2421 q^{14}-13285 q^{13}-17618 q^{12}-10578 q^{11}+1640 q^{10}+13082 q^9+18052 q^8+11361 q^7-915 q^6-12789 q^5-18322 q^4-11965 q^3+311 q^2+12426 q+18379+12426 q^{-1} +311 q^{-2} -11965 q^{-3} -18322 q^{-4} -12789 q^{-5} -915 q^{-6} +11361 q^{-7} +18052 q^{-8} +13082 q^{-9} +1640 q^{-10} -10578 q^{-11} -17618 q^{-12} -13285 q^{-13} -2421 q^{-14} +9543 q^{-15} +16876 q^{-16} +13388 q^{-17} +3344 q^{-18} -8241 q^{-19} -15867 q^{-20} -13308 q^{-21} -4286 q^{-22} +6663 q^{-23} +14476 q^{-24} +12969 q^{-25} +5238 q^{-26} -4870 q^{-27} -12727 q^{-28} -12328 q^{-29} -6052 q^{-30} +2988 q^{-31} +10685 q^{-32} +11294 q^{-33} +6552 q^{-34} -1108 q^{-35} -8380 q^{-36} -9928 q^{-37} -6764 q^{-38} -511 q^{-39} +6132 q^{-40} +8212 q^{-41} +6441 q^{-42} +1784 q^{-43} -3891 q^{-44} -6373 q^{-45} -5828 q^{-46} -2562 q^{-47} +2104 q^{-48} +4537 q^{-49} +4768 q^{-50} +2844 q^{-51} -655 q^{-52} -2860 q^{-53} -3672 q^{-54} -2717 q^{-55} -198 q^{-56} +1547 q^{-57} +2499 q^{-58} +2252 q^{-59} +680 q^{-60} -598 q^{-61} -1545 q^{-62} -1693 q^{-63} -750 q^{-64} +49 q^{-65} +788 q^{-66} +1130 q^{-67} +654 q^{-68} +204 q^{-69} -340 q^{-70} -661 q^{-71} -435 q^{-72} -268 q^{-73} +55 q^{-74} +356 q^{-75} +282 q^{-76} +206 q^{-77} +17 q^{-78} -156 q^{-79} -104 q^{-80} -148 q^{-81} -77 q^{-82} +69 q^{-83} +63 q^{-84} +81 q^{-85} +25 q^{-86} -26 q^{-87} +10 q^{-88} -35 q^{-89} -43 q^{-90} +7 q^{-91} +7 q^{-92} +22 q^{-93} +3 q^{-94} -13 q^{-95} +17 q^{-96} -2 q^{-97} -12 q^{-98} - q^{-100} +7 q^{-101} - q^{-102} -7 q^{-103} +5 q^{-104} +2 q^{-105} -2 q^{-106} - q^{-108} +2 q^{-109} -2 q^{-111} + q^{-112} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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