9 26

9_25

9_27

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 26's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 26 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X7,15,8,14 X17,7,18,6 X9,17,10,16 X15,9,16,8
Gauss code	-1, 4, -3, 1, -2, 7, -6, 9, -8, 3, -4, 2, -5, 6, -9, 8, -7, 5
Dowker-Thistlethwaite code	4 10 12 14 16 2 18 8 6
Conway Notation	[311112]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{11, 4}, {3, 9}, {10, 5}, {4, 6}, {9, 11}, {5, 2}, {8, 3}, {6, 1}, {7, 10}, {2, 8}, {1, 7}]

[edit Notes on presentations of 9 26]

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["9 26"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X7,15,8,14 X17,7,18,6 X9,17,10,16 X15,9,16,8

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -2, 7, -6, 9, -8, 3, -4, 2, -5, 6, -9, 8, -7, 5

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 16 2 18 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[311112]

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]=

[math]\displaystyle{ \textrm{BR}(4,\{1,1,1,-2,1,-2,3,-2,3\}) }[/math]

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]=

{ 4, 9, 4 }

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[{11, 4}, {3, 9}, {10, 5}, {4, 6}, {9, 11}, {5, 2}, {8, 3}, {6, 1}, {7, 10}, {2, 8}, {1, 7}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-9] Hyperbolic Volume 10.5958 A-Polynomial See Data:9 26/A-polynomial

[edit Notes for 9 26's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant 2

[edit Notes for 9 26's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+11 t-13+11 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 47, 2 }
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+5 q^5-7 q^4+8 q^3-8 q^2+7 q-4+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -2 z^4 a^{-4} -z^4+6 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -2 z^2+3 a^{-2} -3 a^{-4} + a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +6 z^7 a^{-3} +3 z^7 a^{-5} +5 z^6 a^{-2} +6 z^6 a^{-4} +4 z^6 a^{-6} +3 z^6+a z^5-6 z^5 a^{-1} -11 z^5 a^{-3} -z^5 a^{-5} +3 z^5 a^{-7} -16 z^4 a^{-2} -14 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} -8 z^4-2 a z^3+3 z^3 a^{-1} +7 z^3 a^{-3} -2 z^3 a^{-5} -4 z^3 a^{-7} +13 z^2 a^{-2} +11 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +5 z^2-z a^{-1} -z a^{-3} +z a^{-5} +z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ -q^6+q^4+1+3 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} -2 q^{-14} + q^{-16} - q^{-18} + q^{-22} }[/math]
The G2 invariant	[math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+5 q^{24}-4 q^{22}-4 q^{20}+16 q^{18}-23 q^{16}+28 q^{14}-23 q^{12}+8 q^{10}+15 q^8-39 q^6+53 q^4-49 q^2+30+2 q^{-2} -31 q^{-4} +51 q^{-6} -49 q^{-8} +35 q^{-10} -5 q^{-12} -23 q^{-14} +35 q^{-16} -28 q^{-18} +6 q^{-20} +25 q^{-22} -40 q^{-24} +44 q^{-26} -23 q^{-28} -10 q^{-30} +46 q^{-32} -73 q^{-34} +76 q^{-36} -53 q^{-38} +12 q^{-40} +33 q^{-42} -67 q^{-44} +78 q^{-46} -62 q^{-48} +28 q^{-50} +5 q^{-52} -38 q^{-54} +44 q^{-56} -31 q^{-58} +4 q^{-60} +21 q^{-62} -33 q^{-64} +26 q^{-66} -4 q^{-68} -25 q^{-70} +44 q^{-72} -50 q^{-74} +40 q^{-76} -15 q^{-78} -15 q^{-80} +38 q^{-82} -46 q^{-84} +44 q^{-86} -27 q^{-88} +9 q^{-90} +8 q^{-92} -21 q^{-94} +24 q^{-96} -19 q^{-98} +13 q^{-100} -4 q^{-102} -2 q^{-104} +4 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_26.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^5+2 q^3-q+3 q^{-1} - q^{-3} + q^{-7} -2 q^{-9} +2 q^{-11} -2 q^{-13} + q^{-15} }[/math]
2	[math]\displaystyle{ q^{16}-2 q^{14}-2 q^{12}+6 q^{10}-2 q^8-7 q^6+10 q^4+2 q^2-12+10 q^{-2} +6 q^{-4} -12 q^{-6} +4 q^{-8} +7 q^{-10} -5 q^{-12} -5 q^{-14} +3 q^{-16} +6 q^{-18} -10 q^{-20} - q^{-22} +14 q^{-24} -9 q^{-26} -5 q^{-28} +12 q^{-30} -5 q^{-32} -5 q^{-34} +6 q^{-36} - q^{-38} -2 q^{-40} + q^{-42} }[/math]
3	[math]\displaystyle{ -q^{33}+2 q^{31}+2 q^{29}-3 q^{27}-6 q^{25}+2 q^{23}+13 q^{21}-q^{19}-19 q^{17}-6 q^{15}+26 q^{13}+16 q^{11}-27 q^9-31 q^7+27 q^5+42 q^3-16 q-51 q^{-1} +9 q^{-3} +56 q^{-5} +4 q^{-7} -54 q^{-9} -15 q^{-11} +50 q^{-13} +19 q^{-15} -38 q^{-17} -28 q^{-19} +25 q^{-21} +30 q^{-23} -8 q^{-25} -32 q^{-27} -9 q^{-29} +30 q^{-31} +29 q^{-33} -27 q^{-35} -43 q^{-37} +20 q^{-39} +54 q^{-41} -10 q^{-43} -57 q^{-45} +53 q^{-49} +9 q^{-51} -45 q^{-53} -14 q^{-55} +32 q^{-57} +14 q^{-59} -21 q^{-61} -12 q^{-63} +13 q^{-65} +10 q^{-67} -8 q^{-69} -5 q^{-71} +3 q^{-73} +3 q^{-75} - q^{-77} -2 q^{-79} + q^{-81} }[/math]
4	[math]\displaystyle{ q^{56}-2 q^{54}-2 q^{52}+3 q^{50}+3 q^{48}+6 q^{46}-9 q^{44}-13 q^{42}+2 q^{40}+10 q^{38}+31 q^{36}-9 q^{34}-41 q^{32}-23 q^{30}+8 q^{28}+81 q^{26}+29 q^{24}-55 q^{22}-85 q^{20}-53 q^{18}+115 q^{16}+119 q^{14}+6 q^{12}-130 q^{10}-173 q^8+63 q^6+186 q^4+138 q^2-83-269 q^{-2} -60 q^{-4} +161 q^{-6} +241 q^{-8} +25 q^{-10} -270 q^{-12} -161 q^{-14} +76 q^{-16} +255 q^{-18} +108 q^{-20} -196 q^{-22} -186 q^{-24} -4 q^{-26} +198 q^{-28} +138 q^{-30} -91 q^{-32} -167 q^{-34} -70 q^{-36} +113 q^{-38} +145 q^{-40} +30 q^{-42} -122 q^{-44} -137 q^{-46} -5 q^{-48} +139 q^{-50} +170 q^{-52} -49 q^{-54} -195 q^{-56} -142 q^{-58} +91 q^{-60} +276 q^{-62} +59 q^{-64} -179 q^{-66} -244 q^{-68} -11 q^{-70} +280 q^{-72} +147 q^{-74} -80 q^{-76} -238 q^{-78} -103 q^{-80} +178 q^{-82} +146 q^{-84} +21 q^{-86} -142 q^{-88} -113 q^{-90} +68 q^{-92} +75 q^{-94} +50 q^{-96} -50 q^{-98} -66 q^{-100} +18 q^{-102} +20 q^{-104} +30 q^{-106} -14 q^{-108} -26 q^{-110} +8 q^{-112} +2 q^{-114} +11 q^{-116} -3 q^{-118} -8 q^{-120} +3 q^{-122} +3 q^{-126} - q^{-128} -2 q^{-130} + q^{-132} }[/math]
5	[math]\displaystyle{ -q^{85}+2 q^{83}+2 q^{81}-3 q^{79}-3 q^{77}-3 q^{75}+q^{73}+9 q^{71}+13 q^{69}-2 q^{67}-20 q^{65}-22 q^{63}-7 q^{61}+25 q^{59}+49 q^{57}+33 q^{55}-33 q^{53}-87 q^{51}-70 q^{49}+12 q^{47}+118 q^{45}+151 q^{43}+44 q^{41}-140 q^{39}-239 q^{37}-149 q^{35}+90 q^{33}+325 q^{31}+318 q^{29}+23 q^{27}-350 q^{25}-498 q^{23}-242 q^{21}+283 q^{19}+651 q^{17}+513 q^{15}-83 q^{13}-709 q^{11}-812 q^9-203 q^7+644 q^5+1030 q^3+571 q-444 q^{-1} -1161 q^{-3} -901 q^{-5} +162 q^{-7} +1138 q^{-9} +1163 q^{-11} +163 q^{-13} -1012 q^{-15} -1308 q^{-17} -440 q^{-19} +807 q^{-21} +1315 q^{-23} +646 q^{-25} -568 q^{-27} -1239 q^{-29} -763 q^{-31} +370 q^{-33} +1078 q^{-35} +786 q^{-37} -173 q^{-39} -913 q^{-41} -771 q^{-43} +43 q^{-45} +737 q^{-47} +723 q^{-49} +97 q^{-51} -572 q^{-53} -695 q^{-55} -228 q^{-57} +398 q^{-59} +683 q^{-61} +398 q^{-63} -215 q^{-65} -672 q^{-67} -599 q^{-69} -17 q^{-71} +649 q^{-73} +829 q^{-75} +281 q^{-77} -581 q^{-79} -1025 q^{-81} -601 q^{-83} +431 q^{-85} +1174 q^{-87} +912 q^{-89} -205 q^{-91} -1204 q^{-93} -1173 q^{-95} -90 q^{-97} +1103 q^{-99} +1337 q^{-101} +391 q^{-103} -880 q^{-105} -1346 q^{-107} -639 q^{-109} +569 q^{-111} +1213 q^{-113} +788 q^{-115} -255 q^{-117} -967 q^{-119} -799 q^{-121} -8 q^{-123} +667 q^{-125} +701 q^{-127} +174 q^{-129} -392 q^{-131} -536 q^{-133} -233 q^{-135} +181 q^{-137} +354 q^{-139} +215 q^{-141} -45 q^{-143} -209 q^{-145} -164 q^{-147} -4 q^{-149} +104 q^{-151} +98 q^{-153} +24 q^{-155} -43 q^{-157} -59 q^{-159} -16 q^{-161} +22 q^{-163} +23 q^{-165} +8 q^{-167} -7 q^{-169} -10 q^{-171} -6 q^{-173} +5 q^{-175} +7 q^{-177} -2 q^{-179} -3 q^{-181} +3 q^{-189} - q^{-191} -2 q^{-193} + q^{-195} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^6+q^4+1+3 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} -2 q^{-14} + q^{-16} - q^{-18} + q^{-22} }[/math]
1,1	[math]\displaystyle{ q^{20}-4 q^{18}+10 q^{16}-22 q^{14}+42 q^{12}-70 q^{10}+100 q^8-140 q^6+177 q^4-196 q^2+208-188 q^{-2} +157 q^{-4} -86 q^{-6} +2 q^{-8} +94 q^{-10} -193 q^{-12} +280 q^{-14} -354 q^{-16} +392 q^{-18} -402 q^{-20} +372 q^{-22} -312 q^{-24} +228 q^{-26} -133 q^{-28} +34 q^{-30} +58 q^{-32} -124 q^{-34} +170 q^{-36} -190 q^{-38} +194 q^{-40} -176 q^{-42} +146 q^{-44} -118 q^{-46} +86 q^{-48} -58 q^{-50} +36 q^{-52} -20 q^{-54} +10 q^{-56} -4 q^{-58} + q^{-60} }[/math]
2,0	[math]\displaystyle{ q^{18}-q^{16}-2 q^{14}+q^{12}+2 q^{10}-2 q^8-4 q^6+4 q^4+6 q^2-2-2 q^{-2} +8 q^{-4} +2 q^{-6} -4 q^{-8} + q^{-10} +5 q^{-12} -2 q^{-14} -4 q^{-16} +2 q^{-18} -2 q^{-20} -7 q^{-22} + q^{-24} +4 q^{-26} -5 q^{-28} - q^{-30} +7 q^{-32} +3 q^{-34} -4 q^{-36} +5 q^{-40} -4 q^{-44} + q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{14}-2 q^{12}+2 q^8-6 q^6+4 q^4+6 q^2-7+6 q^{-2} +9 q^{-4} -10 q^{-6} +3 q^{-8} +8 q^{-10} -6 q^{-12} - q^{-14} +3 q^{-16} -5 q^{-20} -4 q^{-22} +7 q^{-24} -4 q^{-26} -7 q^{-28} +12 q^{-30} - q^{-32} -8 q^{-34} +9 q^{-36} -6 q^{-40} +4 q^{-42} -2 q^{-46} + q^{-48} }[/math]
1,0,0	[math]\displaystyle{ -q^7+q^5-q^3+2 q+3 q^{-3} +2 q^{-7} + q^{-9} -2 q^{-15} -3 q^{-19} + q^{-21} - q^{-23} + q^{-25} + q^{-29} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{16}-q^{14}-q^{12}+q^{10}-2 q^8-3 q^6+3 q^4+3 q^2-3+3 q^{-2} +10 q^{-4} +4 q^{-6} -5 q^{-8} +6 q^{-10} +9 q^{-12} -6 q^{-14} -6 q^{-16} +7 q^{-18} -2 q^{-20} -10 q^{-22} + q^{-24} +2 q^{-26} -7 q^{-28} -3 q^{-30} +7 q^{-32} -6 q^{-36} +5 q^{-38} +7 q^{-40} -5 q^{-42} -3 q^{-44} +6 q^{-46} +2 q^{-48} -4 q^{-50} - q^{-52} +2 q^{-54} -2 q^{-58} + q^{-62} }[/math]
1,0,0,0	[math]\displaystyle{ -q^8+q^6-q^4+q^2+1+3 q^{-4} +3 q^{-8} + q^{-10} +2 q^{-12} -2 q^{-18} -2 q^{-20} - q^{-22} -3 q^{-24} + q^{-26} - q^{-28} + q^{-30} + q^{-32} + q^{-36} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+6 q^8-8 q^6+10 q^4-10 q^2+11-8 q^{-2} +7 q^{-4} -3 q^{-8} +10 q^{-10} -14 q^{-12} +19 q^{-14} -21 q^{-16} +20 q^{-18} -19 q^{-20} +14 q^{-22} -11 q^{-24} +4 q^{-26} + q^{-28} -6 q^{-30} +9 q^{-32} -10 q^{-34} +11 q^{-36} -10 q^{-38} +8 q^{-40} -6 q^{-42} +4 q^{-44} -2 q^{-46} + q^{-48} }[/math]
1,0	[math]\displaystyle{ q^{24}-2 q^{20}-2 q^{18}+2 q^{16}+4 q^{14}-2 q^{12}-7 q^{10}-2 q^8+9 q^6+8 q^4-4 q^2-10+ q^{-2} +12 q^{-4} +8 q^{-6} -8 q^{-8} -9 q^{-10} +3 q^{-12} +10 q^{-14} -8 q^{-18} -2 q^{-20} +7 q^{-22} +3 q^{-24} -7 q^{-26} -5 q^{-28} +5 q^{-30} +6 q^{-32} -5 q^{-34} -8 q^{-36} +2 q^{-38} +9 q^{-40} -10 q^{-44} -4 q^{-46} +10 q^{-48} +9 q^{-50} -5 q^{-52} -11 q^{-54} - q^{-56} +10 q^{-58} +6 q^{-60} -5 q^{-62} -7 q^{-64} +5 q^{-68} +2 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-78} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{18}-2 q^{16}+2 q^{14}-4 q^{12}+5 q^{10}-7 q^8+7 q^6-7 q^4+10 q^2-7+8 q^{-2} -4 q^{-4} +7 q^{-6} - q^{-8} - q^{-10} +6 q^{-12} -4 q^{-14} +12 q^{-16} -13 q^{-18} +14 q^{-20} -15 q^{-22} +16 q^{-24} -18 q^{-26} +10 q^{-28} -15 q^{-30} +9 q^{-32} -7 q^{-34} +2 q^{-36} -2 q^{-38} - q^{-40} +8 q^{-42} -5 q^{-44} +7 q^{-46} -8 q^{-48} +10 q^{-50} -7 q^{-52} +6 q^{-54} -7 q^{-56} +5 q^{-58} -3 q^{-60} +2 q^{-62} -2 q^{-64} + q^{-66} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+5 q^{24}-4 q^{22}-4 q^{20}+16 q^{18}-23 q^{16}+28 q^{14}-23 q^{12}+8 q^{10}+15 q^8-39 q^6+53 q^4-49 q^2+30+2 q^{-2} -31 q^{-4} +51 q^{-6} -49 q^{-8} +35 q^{-10} -5 q^{-12} -23 q^{-14} +35 q^{-16} -28 q^{-18} +6 q^{-20} +25 q^{-22} -40 q^{-24} +44 q^{-26} -23 q^{-28} -10 q^{-30} +46 q^{-32} -73 q^{-34} +76 q^{-36} -53 q^{-38} +12 q^{-40} +33 q^{-42} -67 q^{-44} +78 q^{-46} -62 q^{-48} +28 q^{-50} +5 q^{-52} -38 q^{-54} +44 q^{-56} -31 q^{-58} +4 q^{-60} +21 q^{-62} -33 q^{-64} +26 q^{-66} -4 q^{-68} -25 q^{-70} +44 q^{-72} -50 q^{-74} +40 q^{-76} -15 q^{-78} -15 q^{-80} +38 q^{-82} -46 q^{-84} +44 q^{-86} -27 q^{-88} +9 q^{-90} +8 q^{-92} -21 q^{-94} +24 q^{-96} -19 q^{-98} +13 q^{-100} -4 q^{-102} -2 q^{-104} +4 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/math]

.

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["9 26"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-5 t^2+11 t-13+11 t^{-1} -5 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+z^4+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-3 q^6+5 q^5-7 q^4+8 q^3-8 q^2+7 q-4+3 q^{-1} - q^{-2} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -2 z^4 a^{-4} -z^4+6 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -2 z^2+3 a^{-2} -3 a^{-4} + a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +6 z^7 a^{-3} +3 z^7 a^{-5} +5 z^6 a^{-2} +6 z^6 a^{-4} +4 z^6 a^{-6} +3 z^6+a z^5-6 z^5 a^{-1} -11 z^5 a^{-3} -z^5 a^{-5} +3 z^5 a^{-7} -16 z^4 a^{-2} -14 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} -8 z^4-2 a z^3+3 z^3 a^{-1} +7 z^3 a^{-3} -2 z^3 a^{-5} -4 z^3 a^{-7} +13 z^2 a^{-2} +11 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +5 z^2-z a^{-1} -z a^{-3} +z a^{-5} +z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n25,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

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["9 26"];

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]=

{ [math]\displaystyle{ t^3-5 t^2+11 t-13+11 t^{-1} -5 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ q^7-3 q^6+5 q^5-7 q^4+8 q^3-8 q^2+7 q-4+3 q^{-1} - q^{-2} }[/math] }

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]=

{K11n25,}

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:

(0, -1)

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 [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{176}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ -\frac{88}{3} }[/math] [math]\displaystyle{ -\frac{112}{3} }[/math] [math]\displaystyle{ 0 }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]2 is the signature of 9 26. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 χ 15                   1 1 13                 2   -2 11               3 1   2 9             4 2     -2 7           4 3       1 5         4 4         0 3       3 4           -1 1     2 5             3 -1   1 2               -1 -3   2                 2 -5 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^{20}-3 q^{19}+q^{18}+8 q^{17}-14 q^{16}+q^{15}+25 q^{14}-31 q^{13}-3 q^{12}+48 q^{11}-46 q^{10}-12 q^9+64 q^8-49 q^7-20 q^6+64 q^5-37 q^4-23 q^3+48 q^2-19 q-19+26 q^{-1} -5 q^{-2} -11 q^{-3} +9 q^{-4} -3 q^{-6} + q^{-7} }[/math]
3	[math]\displaystyle{ q^{39}-3 q^{38}+q^{37}+4 q^{36}+q^{35}-11 q^{34}-2 q^{33}+22 q^{32}+4 q^{31}-36 q^{30}-11 q^{29}+57 q^{28}+22 q^{27}-82 q^{26}-42 q^{25}+111 q^{24}+66 q^{23}-135 q^{22}-99 q^{21}+158 q^{20}+130 q^{19}-169 q^{18}-162 q^{17}+174 q^{16}+186 q^{15}-168 q^{14}-201 q^{13}+151 q^{12}+210 q^{11}-130 q^{10}-206 q^9+98 q^8+200 q^7-73 q^6-175 q^5+33 q^4+161 q^3-15 q^2-123 q-14+101 q^{-1} +20 q^{-2} -65 q^{-3} -29 q^{-4} +43 q^{-5} +24 q^{-6} -22 q^{-7} -19 q^{-8} +11 q^{-9} +11 q^{-10} -4 q^{-11} -5 q^{-12} +3 q^{-14} - q^{-15} }[/math]
4	[math]\displaystyle{ q^{64}-3 q^{63}+q^{62}+4 q^{61}-3 q^{60}+4 q^{59}-14 q^{58}+6 q^{57}+18 q^{56}-12 q^{55}+10 q^{54}-48 q^{53}+18 q^{52}+62 q^{51}-22 q^{50}+8 q^{49}-132 q^{48}+34 q^{47}+162 q^{46}+3 q^{45}+q^{44}-313 q^{43}+5 q^{42}+325 q^{41}+128 q^{40}+33 q^{39}-594 q^{38}-130 q^{37}+483 q^{36}+355 q^{35}+166 q^{34}-885 q^{33}-363 q^{32}+548 q^{31}+593 q^{30}+383 q^{29}-1070 q^{28}-596 q^{27}+495 q^{26}+739 q^{25}+602 q^{24}-1101 q^{23}-740 q^{22}+363 q^{21}+754 q^{20}+754 q^{19}-986 q^{18}-772 q^{17}+180 q^{16}+657 q^{15}+830 q^{14}-757 q^{13}-712 q^{12}-22 q^{11}+475 q^{10}+820 q^9-453 q^8-565 q^7-201 q^6+238 q^5+711 q^4-158 q^3-349 q^2-281 q+17+502 q^{-1} +28 q^{-2} -128 q^{-3} -233 q^{-4} -106 q^{-5} +266 q^{-6} +71 q^{-7} +8 q^{-8} -120 q^{-9} -110 q^{-10} +98 q^{-11} +39 q^{-12} +38 q^{-13} -36 q^{-14} -58 q^{-15} +25 q^{-16} +8 q^{-17} +20 q^{-18} -4 q^{-19} -18 q^{-20} +4 q^{-21} +5 q^{-23} -3 q^{-25} + q^{-26} }[/math]
5	[math]\displaystyle{ q^{95}-3 q^{94}+q^{93}+4 q^{92}-3 q^{91}+q^{89}-6 q^{88}+2 q^{87}+13 q^{86}-5 q^{85}-11 q^{84}-3 q^{83}-3 q^{82}+17 q^{81}+28 q^{80}-6 q^{79}-49 q^{78}-46 q^{77}+13 q^{76}+84 q^{75}+102 q^{74}-157 q^{72}-206 q^{71}-32 q^{70}+248 q^{69}+362 q^{68}+139 q^{67}-330 q^{66}-620 q^{65}-335 q^{64}+392 q^{63}+928 q^{62}+666 q^{61}-364 q^{60}-1295 q^{59}-1126 q^{58}+224 q^{57}+1640 q^{56}+1709 q^{55}+61 q^{54}-1939 q^{53}-2334 q^{52}-483 q^{51}+2106 q^{50}+2980 q^{49}+1007 q^{48}-2173 q^{47}-3527 q^{46}-1566 q^{45}+2075 q^{44}+3979 q^{43}+2124 q^{42}-1911 q^{41}-4270 q^{40}-2598 q^{39}+1651 q^{38}+4423 q^{37}+2986 q^{36}-1363 q^{35}-4450 q^{34}-3264 q^{33}+1069 q^{32}+4350 q^{31}+3443 q^{30}-750 q^{29}-4165 q^{28}-3549 q^{27}+443 q^{26}+3883 q^{25}+3566 q^{24}-81 q^{23}-3539 q^{22}-3535 q^{21}-251 q^{20}+3069 q^{19}+3424 q^{18}+659 q^{17}-2580 q^{16}-3243 q^{15}-959 q^{14}+1936 q^{13}+2948 q^{12}+1330 q^{11}-1366 q^{10}-2574 q^9-1467 q^8+689 q^7+2080 q^6+1626 q^5-191 q^4-1574 q^3-1492 q^2-287 q+1017+1366 q^{-1} +526 q^{-2} -559 q^{-3} -1033 q^{-4} -673 q^{-5} +170 q^{-6} +757 q^{-7} +629 q^{-8} +67 q^{-9} -437 q^{-10} -535 q^{-11} -198 q^{-12} +232 q^{-13} +373 q^{-14} +215 q^{-15} -64 q^{-16} -240 q^{-17} -191 q^{-18} -3 q^{-19} +134 q^{-20} +125 q^{-21} +35 q^{-22} -56 q^{-23} -84 q^{-24} -36 q^{-25} +28 q^{-26} +43 q^{-27} +18 q^{-28} -2 q^{-29} -18 q^{-30} -20 q^{-31} +4 q^{-32} +11 q^{-33} +3 q^{-34} -5 q^{-37} +3 q^{-39} - q^{-40} }[/math]
6	[math]\displaystyle{ q^{132}-3 q^{131}+q^{130}+4 q^{129}-3 q^{128}-3 q^{126}+9 q^{125}-10 q^{124}-3 q^{123}+20 q^{122}-15 q^{121}-5 q^{120}-8 q^{119}+35 q^{118}-16 q^{117}-9 q^{116}+48 q^{115}-56 q^{114}-41 q^{113}-29 q^{112}+120 q^{111}+13 q^{110}+21 q^{109}+110 q^{108}-185 q^{107}-202 q^{106}-160 q^{105}+271 q^{104}+184 q^{103}+259 q^{102}+359 q^{101}-404 q^{100}-692 q^{99}-715 q^{98}+260 q^{97}+508 q^{96}+1049 q^{95}+1292 q^{94}-352 q^{93}-1560 q^{92}-2228 q^{91}-642 q^{90}+477 q^{89}+2494 q^{88}+3640 q^{87}+961 q^{86}-2142 q^{85}-4826 q^{84}-3362 q^{83}-1133 q^{82}+3776 q^{81}+7474 q^{80}+4564 q^{79}-1014 q^{78}-7462 q^{77}-7860 q^{76}-5339 q^{75}+3329 q^{74}+11463 q^{73}+10163 q^{72}+2758 q^{71}-8395 q^{70}-12527 q^{69}-11525 q^{68}+327 q^{67}+13706 q^{66}+15848 q^{65}+8319 q^{64}-6842 q^{63}-15408 q^{62}-17574 q^{61}-4251 q^{60}+13448 q^{59}+19636 q^{58}+13584 q^{57}-3723 q^{56}-15842 q^{55}-21632 q^{54}-8541 q^{53}+11510 q^{52}+20960 q^{51}+17032 q^{50}-562 q^{49}-14584 q^{48}-23301 q^{47}-11424 q^{46}+9082 q^{45}+20480 q^{44}+18536 q^{43}+1925 q^{42}-12567 q^{41}-23209 q^{40}-13017 q^{39}+6577 q^{38}+18902 q^{37}+18744 q^{36}+4025 q^{35}-9996 q^{34}-21903 q^{33}-13939 q^{32}+3641 q^{31}+16275 q^{30}+18070 q^{29}+6230 q^{28}-6516 q^{27}-19309 q^{26}-14379 q^{25}-17 q^{24}+12256 q^{23}+16248 q^{22}+8380 q^{21}-2039 q^{20}-15052 q^{19}-13741 q^{18}-3837 q^{17}+6926 q^{16}+12722 q^{15}+9457 q^{14}+2570 q^{13}-9316 q^{12}-11221 q^{11}-6390 q^{10}+1428 q^9+7634 q^8+8378 q^7+5645 q^6-3440 q^5-6936 q^4-6427 q^3-2386 q^2+2408 q+5274+5972 q^{-1} +637 q^{-2} -2421 q^{-3} -4211 q^{-4} -3375 q^{-5} -1043 q^{-6} +1772 q^{-7} +4049 q^{-8} +1940 q^{-9} +469 q^{-10} -1498 q^{-11} -2226 q^{-12} -1966 q^{-13} -383 q^{-14} +1704 q^{-15} +1319 q^{-16} +1203 q^{-17} +109 q^{-18} -710 q^{-19} -1322 q^{-20} -849 q^{-21} +326 q^{-22} +386 q^{-23} +760 q^{-24} +435 q^{-25} +65 q^{-26} -517 q^{-27} -511 q^{-28} -50 q^{-29} -40 q^{-30} +259 q^{-31} +235 q^{-32} +178 q^{-33} -124 q^{-34} -183 q^{-35} -37 q^{-36} -79 q^{-37} +47 q^{-38} +66 q^{-39} +91 q^{-40} -20 q^{-41} -46 q^{-42} -3 q^{-43} -32 q^{-44} +3 q^{-45} +9 q^{-46} +29 q^{-47} -4 q^{-48} -11 q^{-49} +4 q^{-50} -7 q^{-51} +5 q^{-54} -3 q^{-56} + q^{-57} }[/math]
7	[math]\displaystyle{ q^{175}-3 q^{174}+q^{173}+4 q^{172}-3 q^{171}-3 q^{169}+5 q^{168}+5 q^{167}-15 q^{166}+4 q^{165}+10 q^{164}-9 q^{163}+q^{162}-9 q^{161}+18 q^{160}+30 q^{159}-39 q^{158}-4 q^{157}+2 q^{156}-38 q^{155}+8 q^{154}-18 q^{153}+72 q^{152}+128 q^{151}-40 q^{150}-32 q^{149}-97 q^{148}-196 q^{147}-43 q^{146}-29 q^{145}+234 q^{144}+475 q^{143}+166 q^{142}+29 q^{141}-382 q^{140}-777 q^{139}-496 q^{138}-299 q^{137}+523 q^{136}+1397 q^{135}+1170 q^{134}+775 q^{133}-589 q^{132}-2133 q^{131}-2238 q^{130}-1867 q^{129}+209 q^{128}+2976 q^{127}+3962 q^{126}+3859 q^{125}+831 q^{124}-3669 q^{123}-6152 q^{122}-6934 q^{121}-3237 q^{120}+3536 q^{119}+8727 q^{118}+11480 q^{117}+7410 q^{116}-2142 q^{115}-11013 q^{114}-17113 q^{113}-13826 q^{112}-1577 q^{111}+12246 q^{110}+23582 q^{109}+22543 q^{108}+8039 q^{107}-11485 q^{106}-29839 q^{105}-33120 q^{104}-17630 q^{103}+7839 q^{102}+34850 q^{101}+44718 q^{100}+30029 q^{99}-850 q^{98}-37563 q^{97}-56110 q^{96}-44371 q^{95}-9338 q^{94}+37217 q^{93}+65925 q^{92}+59393 q^{91}+22152 q^{90}-33574 q^{89}-73308 q^{88}-73745 q^{87}-36224 q^{86}+27132 q^{85}+77469 q^{84}+86020 q^{83}+50450 q^{82}-18532 q^{81}-78644 q^{80}-95665 q^{79}-63379 q^{78}+9165 q^{77}+77112 q^{76}+102096 q^{75}+74261 q^{74}+242 q^{73}-73763 q^{72}-105780 q^{71}-82631 q^{70}-8634 q^{69}+69412 q^{68}+107020 q^{67}+88485 q^{66}+15723 q^{65}-64638 q^{64}-106546 q^{63}-92286 q^{62}-21426 q^{61}+59967 q^{60}+104969 q^{59}+94376 q^{58}+25954 q^{57}-55370 q^{56}-102562 q^{55}-95408 q^{54}-29849 q^{53}+50781 q^{52}+99604 q^{51}+95724 q^{50}+33410 q^{49}-45807 q^{48}-95872 q^{47}-95516 q^{46}-37197 q^{45}+40027 q^{44}+91288 q^{43}+94774 q^{42}+41244 q^{41}-33105 q^{40}-85283 q^{39}-93303 q^{38}-45740 q^{37}+24934 q^{36}+77726 q^{35}+90575 q^{34}+50115 q^{33}-15322 q^{32}-68123 q^{31}-86413 q^{30}-54154 q^{29}+5066 q^{28}+56812 q^{27}+79873 q^{26}+56737 q^{25}+5750 q^{24}-43635 q^{23}-71394 q^{22}-57635 q^{21}-15435 q^{20}+29857 q^{19}+60242 q^{18}+55565 q^{17}+23803 q^{16}-15745 q^{15}-47752 q^{14}-51047 q^{13}-29078 q^{12}+3342 q^{11}+33994 q^{10}+43390 q^9+31400 q^8+7164 q^7-20876 q^6-34232 q^5-30061 q^4-13954 q^3+9077 q^2+23754 q+26028+17594 q^{-1} -74 q^{-2} -14067 q^{-3} -19912 q^{-4} -17539 q^{-5} -5931 q^{-6} +5636 q^{-7} +13249 q^{-8} +15234 q^{-9} +8625 q^{-10} +251 q^{-11} -6930 q^{-12} -11303 q^{-13} -8874 q^{-14} -3797 q^{-15} +2120 q^{-16} +7236 q^{-17} +7281 q^{-18} +4985 q^{-19} +1075 q^{-20} -3583 q^{-21} -5085 q^{-22} -4758 q^{-23} -2533 q^{-24} +1132 q^{-25} +2876 q^{-26} +3557 q^{-27} +2797 q^{-28} +376 q^{-29} -1181 q^{-30} -2342 q^{-31} -2380 q^{-32} -861 q^{-33} +201 q^{-34} +1209 q^{-35} +1622 q^{-36} +918 q^{-37} +334 q^{-38} -511 q^{-39} -1055 q^{-40} -680 q^{-41} -368 q^{-42} +123 q^{-43} +520 q^{-44} +401 q^{-45} +372 q^{-46} +60 q^{-47} -287 q^{-48} -241 q^{-49} -217 q^{-50} -53 q^{-51} +107 q^{-52} +75 q^{-53} +138 q^{-54} +88 q^{-55} -51 q^{-56} -56 q^{-57} -72 q^{-58} -24 q^{-59} +27 q^{-60} -5 q^{-61} +28 q^{-62} +31 q^{-63} -3 q^{-64} -9 q^{-65} -20 q^{-66} -5 q^{-67} +11 q^{-68} -4 q^{-69} +7 q^{-71} -5 q^{-74} +3 q^{-76} - q^{-77} }[/math]

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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