K11a103

K11a102

K11a104

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a103 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X14,6,15,5 X12,8,13,7 X18,9,19,10 X2,11,3,12 X6,14,7,13 X22,16,1,15 X20,18,21,17 X8,19,9,20 X16,22,17,21
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 Chiral Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a103/ThurstonBennequinNumber Hyperbolic Volume 13.2218 A-Polynomial See Data:K11a103/A-polynomial

[edit Notes for K11a103's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a103's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-20t+33-20t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}-4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 81, 0 }
Jones polynomial	$-q^{7}+3q^{6}-5q^{5}+9q^{4}-11q^{3}+12q^{2}-13q+11-8q^{-1}+5q^{-2}-2q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-2z^{2}a^{2}+z^{4}-2z^{2}-1+2z^{4}a^{-2}+z^{2}a^{-2}+z^{4}a^{-4}+a^{-4}-z^{2}a^{-6}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+5z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}a^{-6}+4z^{8}+4az^{7}-3z^{7}a^{-1}-18z^{7}a^{-3}-10z^{7}a^{-5}+z^{7}a^{-7}+3a^{2}z^{6}-20z^{6}a^{-2}-25z^{6}a^{-4}-13z^{6}a^{-6}-5z^{6}+2a^{3}z^{5}-5az^{5}-2z^{5}a^{-1}+17z^{5}a^{-3}+8z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-2a^{2}z^{4}+19z^{4}a^{-2}+32z^{4}a^{-4}+17z^{4}a^{-6}+z^{4}-2a^{3}z^{3}+5az^{3}+z^{3}a^{-1}-13z^{3}a^{-3}-3z^{3}a^{-5}+4z^{3}a^{-7}-2a^{4}z^{2}-7z^{2}a^{-2}-16z^{2}a^{-4}-7z^{2}a^{-6}+4z^{2}-2az+2za^{-1}+6za^{-3}+2za^{-5}+a^{4}+a^{-4}-1$
The A2 invariant	Data:K11a103/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a103/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a103 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["K11a103"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-20t+33-20t^{-1}+4t^{-2}$

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]=

{ 81, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+5z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}a^{-6}+4z^{8}+4az^{7}-3z^{7}a^{-1}-18z^{7}a^{-3}-10z^{7}a^{-5}+z^{7}a^{-7}+3a^{2}z^{6}-20z^{6}a^{-2}-25z^{6}a^{-4}-13z^{6}a^{-6}-5z^{6}+2a^{3}z^{5}-5az^{5}-2z^{5}a^{-1}+17z^{5}a^{-3}+8z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-2a^{2}z^{4}+19z^{4}a^{-2}+32z^{4}a^{-4}+17z^{4}a^{-6}+z^{4}-2a^{3}z^{3}+5az^{3}+z^{3}a^{-1}-13z^{3}a^{-3}-3z^{3}a^{-5}+4z^{3}a^{-7}-2a^{4}z^{2}-7z^{2}a^{-2}-16z^{2}a^{-4}-7z^{2}a^{-6}+4z^{2}-2az+2za^{-1}+6za^{-3}+2za^{-5}+a^{4}+a^{-4}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a201,}

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["K11a103"];

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]=

{ $4t^{2}-20t+33-20t^{-1}+4t^{-2}$, $-q^{7}+3q^{6}-5q^{5}+9q^{4}-11q^{3}+12q^{2}-13q+11-8q^{-1}+5q^{-2}-2q^{-3}+q^{-4}$ }

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]=

{K11a201,}

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, 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 $-16$ $0$ $128$ ${\frac {424}{3}}$ ${\frac {104}{3}}$ $0$ $-64$ $-64$ $0$ $-{\frac {2048}{3}}$ $0$ $-{\frac {6784}{3}}$ $-{\frac {1664}{3}}$ $-{\frac {31022}{15}}$ $-{\frac {1384}{5}}$ $-{\frac {37448}{45}}$ ${\frac {686}{9}}$ $-{\frac {1742}{15}}$

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=$0 is the signature of K11a103. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a102 K11a104