10 140

10_139

10_141

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 140's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 140 at Knotilus!
	10_140 is also known as the pretzel knot P(4,3,-3).

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,19,12,18 X14,5,15,6 X6,17,7,18 X16,7,17,8 X8,15,9,16 X13,1,14,20 X19,13,20,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, 4, -5, 6, -7, -10, 2, -3, 9, -8, -4, 7, -6, 5, 3, -9, 8
Dowker-Thistlethwaite code	4 10 -14 -16 2 18 20 -8 -6 12
Conway Notation	[4,3,21-]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{9, 2}, {1, 7}, {6, 8}, {7, 9}, {10, 13}, {8, 12}, {13, 11}, {12, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 10}, {11, 1}]

[edit Notes on presentations of 10 140]

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["10 140"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X11,19,12,18 X14,5,15,6 X6,17,7,18 X16,7,17,8 X8,15,9,16 X13,1,14,20 X19,13,20,12 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

-1, 10, -2, 1, 4, -5, 6, -7, -10, 2, -3, 9, -8, -4, 7, -6, 5, 3, -9, 8

In[6]:=

DTCode[K]

Out[6]=

4 10 -14 -16 2 18 20 -8 -6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4,3,21-]

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]=

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

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, 11, 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[{9, 2}, {1, 7}, {6, 8}, {7, 9}, {10, 13}, {8, 12}, {13, 11}, {12, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 10}, {11, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-8][-1] Hyperbolic Volume 5.21257 A-Polynomial See Data:10 140/A-polynomial

[edit Notes for 10 140's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 10 140'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^2-2 t+3-2 t^{-1} + t^{-2} }
Conway polynomial	$z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 9, 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 1- q^{-1} + q^{-2} - q^{-3} +2 q^{-4} - q^{-5} + q^{-6} - q^{-7} }
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+4a^{4}-z^{2}a^{2}-2a^{2}+1$
Kauffman polynomial (db, data sources)	$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-6a^{4}z^{6}-6a^{7}z^{5}-11a^{5}z^{5}-5a^{3}z^{5}+11a^{6}z^{4}+12a^{4}z^{4}+a^{2}z^{4}+10a^{7}z^{3}+16a^{5}z^{3}+6a^{3}z^{3}-8a^{6}z^{2}-12a^{4}z^{2}-4a^{2}z^{2}-4a^{7}z-6a^{5}z-2a^{3}z+2a^{6}+4a^{4}+2a^{2}+1$
The A2 invariant	$-q^{22}-q^{20}-q^{18}+2q^{14}+2q^{12}+2q^{10}-q^{6}-q^{4}+1+q^{-2}$
The G2 invariant	$q^{108}+q^{104}-q^{102}-q^{96}+q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{82}-4q^{80}-4q^{70}+q^{68}+3q^{66}-q^{62}-q^{60}+3q^{58}+5q^{56}+2q^{54}-q^{52}+q^{50}+3q^{48}+4q^{46}-2q^{42}+q^{40}+3q^{38}+q^{36}-q^{34}-2q^{30}+q^{28}-3q^{24}-q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-q^{8}+q^{6}+2+q^{-4}+q^{-6}+q^{-10}$

Further Quantum Invariants

Further quantum knot invariants for 10_140.

A1 Invariants.

Weight	Invariant
1	$-q^{15}+q^{9}+q^{7}+q^{-1}$
2	$q^{44}-q^{40}-q^{34}-q^{32}+q^{28}+q^{26}+q^{22}-q^{18}-q^{14}-q^{12}+q^{10}+q^{8}+q^{6}+q^{2}+2-q^{-4}$
3	$-q^{87}+q^{83}+q^{81}-q^{77}+q^{73}+q^{71}-q^{67}-2q^{65}-q^{63}+q^{59}-q^{55}+2q^{51}+2q^{49}-q^{45}+q^{41}-2q^{37}-2q^{35}-q^{29}+q^{25}+q^{23}+q^{17}+q^{15}-q^{11}-q^{9}+2q^{7}+3q^{5}-2q+3q^{-3}+q^{-5}-q^{-7}-q^{-9}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{22}-q^{20}-q^{18}+2q^{14}+2q^{12}+2q^{10}-q^{6}-q^{4}+1+q^{-2}$
1,1	$q^{60}+2q^{56}-2q^{54}+2q^{52}-2q^{50}-2q^{46}-4q^{44}-4q^{40}+2q^{38}+q^{36}+4q^{34}+4q^{32}+4q^{30}+q^{28}-2q^{26}-4q^{24}-4q^{22}-4q^{20}-2q^{18}+6q^{14}+2q^{12}+6q^{10}+2q^{8}+2q^{4}-2q^{2}+2-2q^{-2}+q^{-4}$
2,0	$q^{58}+q^{56}+q^{54}-q^{48}-2q^{46}-4q^{44}-4q^{42}-3q^{40}+4q^{36}+4q^{34}+6q^{32}+4q^{30}+3q^{28}-2q^{26}-4q^{24}-5q^{22}-5q^{20}-3q^{18}+4q^{14}+4q^{12}+5q^{10}+2q^{8}+1-q^{-2}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{46}+q^{42}-q^{38}-2q^{36}-2q^{34}-2q^{32}-3q^{30}+2q^{24}+2q^{22}+4q^{20}+3q^{18}+2q^{16}+2q^{14}-q^{10}-2q^{8}-q^{6}-q^{4}+2+q^{-2}+q^{-4}$
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^{29}-q^{27}-2 q^{25}-q^{23}+2 q^{19}+3 q^{17}+3 q^{15}+2 q^{13}-q^9-2 q^7-q^5+q+ q^{-1} + q^{-3} }

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{60}+q^{58}+2q^{56}+2q^{54}+q^{52}-q^{50}-3q^{48}-6q^{46}-7q^{44}-6q^{42}-4q^{40}-q^{38}+2q^{36}+6q^{34}+6q^{32}+5q^{30}+4q^{28}+3q^{26}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-2q^{10}-2q^{8}-q^{6}-q^{4}+2+2q^{-2}+q^{-4}+q^{-6}$
1,0,0,0	$-q^{36}-q^{34}-2q^{32}-2q^{30}-q^{28}+2q^{24}+3q^{22}+4q^{20}+3q^{18}+2q^{16}-q^{12}-2q^{10}-2q^{8}-q^{6}+q^{2}+1+q^{-2}+q^{-4}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{46}-q^{42}-q^{38}+q^{30}+2q^{26}+2q^{22}+q^{18}-q^{10}-q^{6}+q^{4}+q^{-2}+q^{-4}$
1,0	$q^{76}+q^{68}-q^{64}-q^{62}-q^{58}-2q^{56}-q^{54}-q^{48}+q^{42}+q^{38}+q^{34}+q^{32}+q^{30}+q^{26}+2q^{24}+q^{22}+q^{16}-q^{12}-q^{10}-q^{4}+1+q^{-2}+q^{-6}$

D4 Invariants.

Weight

Invariant

1,0,0,0

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^{62}+q^{58}+q^{54}-q^{52}-q^{50}-2 q^{48}-3 q^{46}-3 q^{44}-4 q^{42}-2 q^{40}-2 q^{38}+q^{36}+q^{34}+4 q^{32}+4 q^{30}+6 q^{28}+4 q^{26}+4 q^{24}+2 q^{22}+q^{20}-2 q^{16}-2 q^{14}-3 q^{12}-q^{10}-2 q^8+q^2+2+ q^{-2} + q^{-4} + q^{-6} }

G2 Invariants.

Weight	Invariant
1,0	$q^{108}+q^{104}-q^{102}-q^{96}+q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{82}-4q^{80}-4q^{70}+q^{68}+3q^{66}-q^{62}-q^{60}+3q^{58}+5q^{56}+2q^{54}-q^{52}+q^{50}+3q^{48}+4q^{46}-2q^{42}+q^{40}+3q^{38}+q^{36}-q^{34}-2q^{30}+q^{28}-3q^{24}-q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-q^{8}+q^{6}+2+q^{-4}+q^{-6}+q^{-10}$

.

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["10 140"];

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^2-2 t+3-2 t^{-1} + t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 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 1- q^{-1} + q^{-2} - q^{-3} +2 q^{-4} - q^{-5} + q^{-6} - q^{-7} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+4a^{4}-z^{2}a^{2}-2a^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-6a^{4}z^{6}-6a^{7}z^{5}-11a^{5}z^{5}-5a^{3}z^{5}+11a^{6}z^{4}+12a^{4}z^{4}+a^{2}z^{4}+10a^{7}z^{3}+16a^{5}z^{3}+6a^{3}z^{3}-8a^{6}z^{2}-12a^{4}z^{2}-4a^{2}z^{2}-4a^{7}z-6a^{5}z-2a^{3}z+2a^{6}+4a^{4}+2a^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_20, K11n73, K11n74,}

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["10 140"];

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]=

{ $t^{2}-2t+3-2t^{-1}+t^{-2}$, $1-q^{-1}+q^{-2}-q^{-3}+2q^{-4}-q^{-5}+q^{-6}-q^{-7}$ }

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]=

{8_20, K11n73, K11n74,}

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:

(2, -4)

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 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 8} $-32$ $32$ 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 \frac{316}{3}} ${\frac {20}{3}}$ $-256$ $-{\frac {1472}{3}}$ $-{\frac {128}{3}}$ $-96$ ${\frac {256}{3}}$ $512$ ${\frac {2528}{3}}$ ${\frac {160}{3}}$ ${\frac {33391}{15}}$ $-{\frac {2444}{15}}$ ${\frac {43564}{45}}$ ${\frac {881}{9}}$ ${\frac {1711}{15}}$

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 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 \chi} (fixed 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} , 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 10 140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$-q+1+2q^{-1}-2q^{-2}+3q^{-4}-2q^{-5}+q^{-7}-2q^{-8}+q^{-9}-q^{-11}+2q^{-12}-q^{-13}+2q^{-15}-2q^{-16}-q^{-17}+2q^{-18}-q^{-19}-q^{-20}+q^{-21}$
3	$-q^{3}+2q+2-4q^{-1}-2q^{-2}+4q^{-3}+5q^{-4}-5q^{-5}-5q^{-6}+4q^{-7}+6q^{-8}-4q^{-9}-5q^{-10}+3q^{-11}+6q^{-12}-3q^{-13}-5q^{-14}+2q^{-15}+5q^{-16}-2q^{-17}-5q^{-18}+5q^{-20}-4q^{-22}-q^{-23}+4q^{-24}+q^{-25}-2q^{-26}-q^{-27}+2q^{-28}-q^{-30}+q^{-32}-q^{-33}-2q^{-34}+q^{-35}+2q^{-36}-2q^{-38}+q^{-40}+q^{-41}-q^{-42}$
4	$q^{3}-2q^{2}-2q+4q^{-1}+7q^{-2}-5q^{-3}-6q^{-4}-4q^{-5}+6q^{-6}+12q^{-7}-4q^{-8}-7q^{-9}-8q^{-10}+4q^{-11}+14q^{-12}-3q^{-13}-6q^{-14}-7q^{-15}+3q^{-16}+11q^{-17}-4q^{-18}-4q^{-19}-5q^{-20}+3q^{-21}+9q^{-22}-3q^{-23}-2q^{-24}-5q^{-25}+q^{-26}+7q^{-27}-q^{-28}-5q^{-30}-2q^{-31}+4q^{-32}+3q^{-34}-2q^{-35}-4q^{-36}+5q^{-39}+2q^{-40}-3q^{-41}-3q^{-42}-2q^{-43}+3q^{-44}+5q^{-45}-3q^{-47}-3q^{-48}-q^{-49}+4q^{-50}-q^{-53}-2q^{-54}+3q^{-55}-2q^{-56}+4q^{-60}-2q^{-61}-q^{-62}-q^{-63}-q^{-64}+3q^{-65}-q^{-68}-q^{-69}+q^{-70}$

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.

10_139 10_141