K11a31

K11a30

K11a32

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a31 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,6,15,5 X2837 X16,10,17,9 X18,11,19,12 X6,14,7,13 X22,16,1,15 X20,18,21,17 X10,19,11,20 X12,22,13,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -11, 7, -3, 8, -5, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code	4 8 14 2 16 18 6 22 20 10 12

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a31/ThurstonBennequinNumber Hyperbolic Volume 15.4016 A-Polynomial See Data:K11a31/A-polynomial

[edit Notes for K11a31's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for K11a31's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{3}+14t^{2}-28t+35-28t^{-1}+14t^{-2}-3t^{-3}$
Conway polynomial	$-3z^{6}-4z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 125, 4 }
Jones polynomial	$-q^{11}+4q^{10}-8q^{9}+13q^{8}-18q^{7}+20q^{6}-20q^{5}+17q^{4}-12q^{3}+8q^{2}-3q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+2z^{2}a^{-2}+3z^{2}a^{-4}-10z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}+a^{-2}+3a^{-4}-6a^{-6}+4a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+8z^{9}a^{-7}+4z^{9}a^{-9}+5z^{8}a^{-4}+13z^{8}a^{-6}+15z^{8}a^{-8}+7z^{8}a^{-10}+3z^{7}a^{-3}-2z^{7}a^{-5}-4z^{7}a^{-7}+8z^{7}a^{-9}+7z^{7}a^{-11}+z^{6}a^{-2}-11z^{6}a^{-4}-37z^{6}a^{-6}-35z^{6}a^{-8}-6z^{6}a^{-10}+4z^{6}a^{-12}-7z^{5}a^{-3}-10z^{5}a^{-5}-20z^{5}a^{-7}-29z^{5}a^{-9}-11z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+8z^{4}a^{-4}+38z^{4}a^{-6}+30z^{4}a^{-8}-3z^{4}a^{-10}-6z^{4}a^{-12}+4z^{3}a^{-3}+10z^{3}a^{-5}+25z^{3}a^{-7}+26z^{3}a^{-9}+6z^{3}a^{-11}-z^{3}a^{-13}+3z^{2}a^{-2}-6z^{2}a^{-4}-22z^{2}a^{-6}-13z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}a^{-12}-4za^{-5}-9za^{-7}-7za^{-9}-2za^{-11}-a^{-2}+3a^{-4}+6a^{-6}+4a^{-8}+a^{-10}$
The A2 invariant	Data:K11a31/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a31/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a31 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["K11a31"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{3}+14t^{2}-28t+35-28t^{-1}+14t^{-2}-3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{6}-4z^{4}+z^{2}+1$

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]=

{ 125, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+4q^{10}-8q^{9}+13q^{8}-18q^{7}+20q^{6}-20q^{5}+17q^{4}-12q^{3}+8q^{2}-3q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+2z^{2}a^{-2}+3z^{2}a^{-4}-10z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}+a^{-2}+3a^{-4}-6a^{-6}+4a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+8z^{9}a^{-7}+4z^{9}a^{-9}+5z^{8}a^{-4}+13z^{8}a^{-6}+15z^{8}a^{-8}+7z^{8}a^{-10}+3z^{7}a^{-3}-2z^{7}a^{-5}-4z^{7}a^{-7}+8z^{7}a^{-9}+7z^{7}a^{-11}+z^{6}a^{-2}-11z^{6}a^{-4}-37z^{6}a^{-6}-35z^{6}a^{-8}-6z^{6}a^{-10}+4z^{6}a^{-12}-7z^{5}a^{-3}-10z^{5}a^{-5}-20z^{5}a^{-7}-29z^{5}a^{-9}-11z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+8z^{4}a^{-4}+38z^{4}a^{-6}+30z^{4}a^{-8}-3z^{4}a^{-10}-6z^{4}a^{-12}+4z^{3}a^{-3}+10z^{3}a^{-5}+25z^{3}a^{-7}+26z^{3}a^{-9}+6z^{3}a^{-11}-z^{3}a^{-13}+3z^{2}a^{-2}-6z^{2}a^{-4}-22z^{2}a^{-6}-13z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}a^{-12}-4za^{-5}-9za^{-7}-7za^{-9}-2za^{-11}-a^{-2}+3a^{-4}+6a^{-6}+4a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a317,}

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["K11a31"];

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]=

{ $-3t^{3}+14t^{2}-28t+35-28t^{-1}+14t^{-2}-3t^{-3}$, $-q^{11}+4q^{10}-8q^{9}+13q^{8}-18q^{7}+20q^{6}-20q^{5}+17q^{4}-12q^{3}+8q^{2}-3q+1$ }

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]=

{K11a317,}

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, 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 $4$ $8$ $8$ ${\frac {206}{3}}$ ${\frac {106}{3}}$ $32$ ${\frac {1136}{3}}$ ${\frac {320}{3}}$ $136$ ${\frac {32}{3}}$ $32$ ${\frac {824}{3}}$ ${\frac {424}{3}}$ ${\frac {44191}{30}}$ $-{\frac {2182}{15}}$ ${\frac {47942}{45}}$ ${\frac {1697}{18}}$ ${\frac {3391}{30}}$

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=$4 is the signature of K11a31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 23                       1 -1 21                     3   3 19                   5 1   -4 17                 8 3     5 15               10 5       -5 13             10 8         2 11           10 10           0 9         7 10             -3 7       5 10               5 5     3 7                 -4 3   1 6                   5 1   2                     -2 -1 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a30 K11a32