K11a351

K11a350

K11a352

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a351 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X20,9,21,10 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X8,19,9,20 X10,21,11,22
Gauss code	1, -7, 2, -6, 3, -1, 4, -10, 5, -11, 6, -9, 7, -4, 8, -3, 9, -2, 10, -5, 11, -8
Dowker-Thistlethwaite code	6 18 16 14 20 4 2 22 12 8 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a351/ThurstonBennequinNumber Hyperbolic Volume 18.5253 A-Polynomial See Data:K11a351/A-polynomial

[edit Notes for K11a351's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a351's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-7t^{3}+20t^{2}-34t+41-34t^{-1}+20t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 165, 0 }
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-16q^{3}+23q^{2}-26q+27-24q^{-1}+18q^{-2}-11q^{-3}+5q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+5z^{4}-4z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-2}+4z^{10}+11az^{9}+20z^{9}a^{-1}+9z^{9}a^{-3}+14a^{2}z^{8}+8z^{8}a^{-2}+8z^{8}a^{-4}+14z^{8}+11a^{3}z^{7}-10az^{7}-45z^{7}a^{-1}-20z^{7}a^{-3}+4z^{7}a^{-5}+5a^{4}z^{6}-19a^{2}z^{6}-37z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-40z^{6}+a^{5}z^{5}-14a^{3}z^{5}-3az^{5}+37z^{5}a^{-1}+16z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+4a^{2}z^{4}+38z^{4}a^{-2}+17z^{4}a^{-4}-2z^{4}a^{-6}+27z^{4}+3a^{3}z^{3}-az^{3}-13z^{3}a^{-1}-5z^{3}a^{-3}+4z^{3}a^{-5}+a^{2}z^{2}-11z^{2}a^{-2}-6z^{2}a^{-4}-4z^{2}+a^{3}z+2az+2za^{-1}+za^{-3}-a^{2}-a^{-2}-1$
The A2 invariant	$-q^{14}+3q^{12}-3q^{10}+3q^{8}+q^{6}-4q^{4}+5q^{2}-6+4q^{-2}-q^{-4}+5q^{-8}-4q^{-10}+2q^{-12}-q^{-14}-q^{-16}+q^{-18}$
The G2 invariant	Data:K11a351/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a351 Quantum Invariants.

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["K11a351"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-7t^{3}+20t^{2}-34t+41-34t^{-1}+20t^{-2}-7t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+z^{6}-2z^{4}-z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 165, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+9q^{4}-16q^{3}+23q^{2}-26q+27-24q^{-1}+18q^{-2}-11q^{-3}+5q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+5z^{4}-4z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}+a^{2}+a^{-2}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-2}+4z^{10}+11az^{9}+20z^{9}a^{-1}+9z^{9}a^{-3}+14a^{2}z^{8}+8z^{8}a^{-2}+8z^{8}a^{-4}+14z^{8}+11a^{3}z^{7}-10az^{7}-45z^{7}a^{-1}-20z^{7}a^{-3}+4z^{7}a^{-5}+5a^{4}z^{6}-19a^{2}z^{6}-37z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-40z^{6}+a^{5}z^{5}-14a^{3}z^{5}-3az^{5}+37z^{5}a^{-1}+16z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+4a^{2}z^{4}+38z^{4}a^{-2}+17z^{4}a^{-4}-2z^{4}a^{-6}+27z^{4}+3a^{3}z^{3}-az^{3}-13z^{3}a^{-1}-5z^{3}a^{-3}+4z^{3}a^{-5}+a^{2}z^{2}-11z^{2}a^{-2}-6z^{2}a^{-4}-4z^{2}+a^{3}z+2az+2za^{-1}+za^{-3}-a^{2}-a^{-2}-1$

"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["K11a351"];

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]=

{ $t^{4}-7t^{3}+20t^{2}-34t+41-34t^{-1}+20t^{-2}-7t^{-3}+t^{-4}$, $q^{6}-4q^{5}+9q^{4}-16q^{3}+23q^{2}-26q+27-24q^{-1}+18q^{-2}-11q^{-3}+5q^{-4}-q^{-5}$ }

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:

(-1, 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 $-4$ $0$ $8$ ${\frac {82}{3}}$ ${\frac {62}{3}}$ $0$ $0$ $64$ $-64$ $-{\frac {32}{3}}$ $0$ $-{\frac {328}{3}}$ $-{\frac {248}{3}}$ $-{\frac {2911}{30}}$ ${\frac {1262}{15}}$ $-{\frac {8462}{45}}$ ${\frac {415}{18}}$ $-{\frac {991}{30}}$

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 K11a351. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   6 1   5 7                 10 3     -7 5               13 6       7 3             13 10         -3 1           14 13           1 -1         11 14             3 -3       7 13               -6 -5     4 11                 7 -7   1 7                   -6 -9   4                     4 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a350 K11a352