K11n102

K11n101

K11n103

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n102 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,6,14,7 X15,22,16,1 X17,20,18,21 X19,9,20,8 X21,16,22,17
Gauss code	1, -6, 2, -1, -3, 7, -4, 10, -5, -2, 6, 4, -7, 3, -8, 11, -9, 5, -10, 9, -11, 8
Dowker-Thistlethwaite code	4 10 -14 -12 -18 2 -6 -22 -20 -8 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n102/ThurstonBennequinNumber Hyperbolic Volume 7.24432 A-Polynomial See Data:K11n102/A-polynomial

[edit Notes for K11n102'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 K11n102's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+t+1+t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, -2 }
Jones polynomial	$q-1+2q^{-1}-q^{-2}+q^{-3}-q^{-4}-q^{-5}+q^{-6}-q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$a^{8}-z^{2}a^{6}-a^{6}-z^{4}a^{2}-3z^{2}a^{2}-a^{2}+z^{2}+2$
Kauffman polynomial (db, data sources)	$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+a^{4}z^{8}-7a^{7}z^{7}-7a^{5}z^{7}-7a^{8}z^{6}-14a^{6}z^{6}-7a^{4}z^{6}+15a^{7}z^{5}+14a^{5}z^{5}+az^{5}+15a^{8}z^{4}+27a^{6}z^{4}+13a^{4}z^{4}+2a^{2}z^{4}+z^{4}-13a^{7}z^{3}-11a^{5}z^{3}-2az^{3}-11a^{8}z^{2}-15a^{6}z^{2}-6a^{4}z^{2}-5a^{2}z^{2}-3z^{2}+5a^{7}z+6a^{5}z-az+a^{8}+a^{6}+a^{2}+2$
The A2 invariant	Data:K11n102/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n102/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n102 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["K11n102"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+a^{4}z^{8}-7a^{7}z^{7}-7a^{5}z^{7}-7a^{8}z^{6}-14a^{6}z^{6}-7a^{4}z^{6}+15a^{7}z^{5}+14a^{5}z^{5}+az^{5}+15a^{8}z^{4}+27a^{6}z^{4}+13a^{4}z^{4}+2a^{2}z^{4}+z^{4}-13a^{7}z^{3}-11a^{5}z^{3}-2az^{3}-11a^{8}z^{2}-15a^{6}z^{2}-6a^{4}z^{2}-5a^{2}z^{2}-3z^{2}+5a^{7}z+6a^{5}z-az+a^{8}+a^{6}+a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n38,}

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["K11n102"];

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]=

{ $-t^{2}+t+1+t^{-1}-t^{-2}$, $q-1+2q^{-1}-q^{-2}+q^{-3}-q^{-4}-q^{-5}+q^{-6}-q^{-7}+q^{-8}$ }

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]=

{K11n38,}

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:

(-3, 6)

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 $-12$ $48$ $72$ $2$ $46$ $-576$ $-800$ $-192$ $-240$ $-288$ $1152$ $-24$ $-552$ ${\frac {29649}{10}}$ ${\frac {542}{5}}$ ${\frac {4166}{5}}$ ${\frac {709}{2}}$ $-{\frac {591}{10}}$

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 K11n102. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n101 K11n103