K11a311

K11a310

K11a312

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a311 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a311/ThurstonBennequinNumber Hyperbolic Volume 11.8941 A-Polynomial See Data:K11a311/A-polynomial

[edit Notes for K11a311's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for K11a311's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t^{2}+20t-31+20t^{-1}-4t^{-2}$
Conway polynomial	$-4z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 79, 2 }
Jones polynomial	$-q^{10}+2q^{9}-4q^{8}+7q^{7}-10q^{6}+12q^{5}-12q^{4}+12q^{3}-9q^{2}+6q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}a^{-6}+z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}+2z^{2}a^{-8}+z^{2}+a^{-2}+a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+3z^{9}a^{-5}+5z^{9}a^{-7}+2z^{9}a^{-9}+5z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}a^{-10}+6z^{7}a^{-3}-3z^{7}a^{-5}-17z^{7}a^{-7}-7z^{7}a^{-9}+z^{7}a^{-11}+5z^{6}a^{-2}-7z^{6}a^{-4}-11z^{6}a^{-6}-8z^{6}a^{-8}-9z^{6}a^{-10}+3z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}+25z^{5}a^{-7}+7z^{5}a^{-9}-5z^{5}a^{-11}-6z^{4}a^{-2}+3z^{4}a^{-4}+9z^{4}a^{-6}+11z^{4}a^{-8}+12z^{4}a^{-10}+z^{4}-3z^{3}a^{-1}+6z^{3}a^{-3}-3z^{3}a^{-5}-23z^{3}a^{-7}-4z^{3}a^{-9}+7z^{3}a^{-11}+3z^{2}a^{-2}+2z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}a^{-8}-5z^{2}a^{-10}-z^{2}+2za^{-5}+6za^{-7}+2za^{-9}-2za^{-11}-a^{-2}+a^{-8}+a^{-10}$
The A2 invariant	Data:K11a311/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a311/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a311 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["K11a311"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t^{2}+20t-31+20t^{-1}-4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 79, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{10}+2q^{9}-4q^{8}+7q^{7}-10q^{6}+12q^{5}-12q^{4}+12q^{3}-9q^{2}+6q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+3z^{9}a^{-5}+5z^{9}a^{-7}+2z^{9}a^{-9}+5z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}a^{-10}+6z^{7}a^{-3}-3z^{7}a^{-5}-17z^{7}a^{-7}-7z^{7}a^{-9}+z^{7}a^{-11}+5z^{6}a^{-2}-7z^{6}a^{-4}-11z^{6}a^{-6}-8z^{6}a^{-8}-9z^{6}a^{-10}+3z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}+25z^{5}a^{-7}+7z^{5}a^{-9}-5z^{5}a^{-11}-6z^{4}a^{-2}+3z^{4}a^{-4}+9z^{4}a^{-6}+11z^{4}a^{-8}+12z^{4}a^{-10}+z^{4}-3z^{3}a^{-1}+6z^{3}a^{-3}-3z^{3}a^{-5}-23z^{3}a^{-7}-4z^{3}a^{-9}+7z^{3}a^{-11}+3z^{2}a^{-2}+2z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}a^{-8}-5z^{2}a^{-10}-z^{2}+2za^{-5}+6za^{-7}+2za^{-9}-2za^{-11}-a^{-2}+a^{-8}+a^{-10}$

"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["K11a311"];

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]=

{ $-4t^{2}+20t-31+20t^{-1}-4t^{-2}$, $-q^{10}+2q^{9}-4q^{8}+7q^{7}-10q^{6}+12q^{5}-12q^{4}+12q^{3}-9q^{2}+6q-3+q^{-1}$ }

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]=

{}

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:

(4, 10)

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 $16$ $80$ $128$ ${\frac {1400}{3}}$ ${\frac {280}{3}}$ $1280$ ${\frac {8768}{3}}$ ${\frac {1280}{3}}$ $656$ ${\frac {2048}{3}}$ $3200$ ${\frac {22400}{3}}$ ${\frac {4480}{3}}$ ${\frac {272702}{15}}$ $-{\frac {31048}{15}}$ ${\frac {463208}{45}}$ ${\frac {3970}{9}}$ ${\frac {23582}{15}}$

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 $\chi$ (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=$2 is the signature of K11a311. 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 21                       1 -1 19                     1   1 17                   3 1   -2 15                 4 1     3 13               6 3       -3 11             6 4         2 9           6 6           0 7         6 6             0 5       3 6               3 3     3 6                 -3 1   1 4                   3 -1   2                     -2 -3 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a310 K11a312