K11n31

K11n30

K11n32

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n31 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,6,13,5 X2837 X9,17,10,16 X6,12,7,11 X13,20,14,21 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,19,22,18
Gauss code	1, -4, 2, -1, 3, -6, 4, -2, -5, 8, 6, -3, -7, 10, -8, 5, -9, 11, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 8 12 2 -16 6 -20 -10 -22 -14 -18

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n31/ThurstonBennequinNumber Hyperbolic Volume 10.5314 A-Polynomial See Data:K11n31/A-polynomial

[edit Notes for K11n31'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 K11n31's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+2t^{2}+2t-5+2t^{-1}+2t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-4z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, 2 }
Jones polynomial	$-q^{10}+2q^{9}-2q^{8}+2q^{7}-q^{6}-q^{3}+2q^{2}-q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+4z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}+2z^{2}a^{-8}+3a^{-2}-3a^{-4}+2a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-7}+z^{9}a^{-9}+z^{8}a^{-6}+3z^{8}a^{-8}+2z^{8}a^{-10}+z^{7}a^{-3}-6z^{7}a^{-7}-4z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-2}+2z^{6}a^{-4}-7z^{6}a^{-6}-19z^{6}a^{-8}-11z^{6}a^{-10}-4z^{5}a^{-3}+8z^{5}a^{-7}-z^{5}a^{-9}-5z^{5}a^{-11}-5z^{4}a^{-2}-10z^{4}a^{-4}+11z^{4}a^{-6}+32z^{4}a^{-8}+16z^{4}a^{-10}+2z^{3}a^{-3}-4z^{3}a^{-5}-4z^{3}a^{-7}+8z^{3}a^{-9}+6z^{3}a^{-11}+7z^{2}a^{-2}+10z^{2}a^{-4}-6z^{2}a^{-6}-17z^{2}a^{-8}-8z^{2}a^{-10}+za^{-3}+3za^{-5}+2za^{-7}-2za^{-9}-2za^{-11}-3a^{-2}-3a^{-4}+2a^{-8}+a^{-10}$
The A2 invariant	Data:K11n31/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n31/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n31 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["K11n31"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+2t^{2}+2t-5+2t^{-1}+2t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 3, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+4z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}+2z^{2}a^{-8}+3a^{-2}-3a^{-4}+2a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-7}+z^{9}a^{-9}+z^{8}a^{-6}+3z^{8}a^{-8}+2z^{8}a^{-10}+z^{7}a^{-3}-6z^{7}a^{-7}-4z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-2}+2z^{6}a^{-4}-7z^{6}a^{-6}-19z^{6}a^{-8}-11z^{6}a^{-10}-4z^{5}a^{-3}+8z^{5}a^{-7}-z^{5}a^{-9}-5z^{5}a^{-11}-5z^{4}a^{-2}-10z^{4}a^{-4}+11z^{4}a^{-6}+32z^{4}a^{-8}+16z^{4}a^{-10}+2z^{3}a^{-3}-4z^{3}a^{-5}-4z^{3}a^{-7}+8z^{3}a^{-9}+6z^{3}a^{-11}+7z^{2}a^{-2}+10z^{2}a^{-4}-6z^{2}a^{-6}-17z^{2}a^{-8}-8z^{2}a^{-10}+za^{-3}+3za^{-5}+2za^{-7}-2za^{-9}-2za^{-11}-3a^{-2}-3a^{-4}+2a^{-8}+a^{-10}$

"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["K11n31"];

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]=

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

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:

(1, 3)

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 $4$ $24$ $8$ ${\frac {446}{3}}$ ${\frac {106}{3}}$ $96$ $656$ $128$ $120$ ${\frac {32}{3}}$ $288$ ${\frac {1784}{3}}$ ${\frac {424}{3}}$ ${\frac {79951}{30}}$ $-{\frac {34}{5}}$ ${\frac {54422}{45}}$ ${\frac {977}{18}}$ ${\frac {4111}{30}}$

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=$2 is the signature of K11n31. 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                   1 1   0 15               1 2 1     0 13             1 1 1       1 11           1 2 2         -1 9         1 2 1           0 7       1 1 1             -1 5     1 1 1               1 3   1 2                   1 1                         0 -1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11n30 K11n32