K11n103

K11n102

K11n104

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n103 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X14,5,15,6 X12,8,13,7 X9,20,10,21 X2,11,3,12 X16,13,17,14 X6,15,7,16 X17,22,18,1 X19,8,20,9 X21,18,22,19
Gauss code	1, -6, 2, -1, 3, -8, 4, 10, -5, -2, 6, -4, 7, -3, 8, -7, -9, 11, -10, 5, -11, 9
Dowker-Thistlethwaite code	4 10 14 12 -20 2 16 6 -22 -8 -18

A Braid Representative
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:K11n103/ThurstonBennequinNumber Hyperbolic Volume 13.4162 A-Polynomial See Data:K11n103/A-polynomial

[edit Notes for K11n103'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 K11n103's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-15t+19-15t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 65, -4 }
Jones polynomial	$1-3q^{-1}+6q^{-2}-8q^{-3}+11q^{-4}-11q^{-5}+10q^{-6}-8q^{-7}+5q^{-8}-2q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$-2z^{2}a^{8}-2a^{8}+3z^{4}a^{6}+7z^{2}a^{6}+3a^{6}-z^{6}a^{4}-3z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$3z^{3}a^{11}-2za^{11}+z^{6}a^{10}+4z^{4}a^{10}-3z^{2}a^{10}+3z^{7}a^{9}-2z^{5}a^{9}+3z^{3}a^{9}-za^{9}+3z^{8}a^{8}-2z^{6}a^{8}+3z^{2}a^{8}-2a^{8}+z^{9}a^{7}+6z^{7}a^{7}-14z^{5}a^{7}+7z^{3}a^{7}-za^{7}+6z^{8}a^{6}-9z^{6}a^{6}-5z^{4}a^{6}+9z^{2}a^{6}-3a^{6}+z^{9}a^{5}+6z^{7}a^{5}-21z^{5}a^{5}+14z^{3}a^{5}-3za^{5}+3z^{8}a^{4}-5z^{6}a^{4}-4z^{4}a^{4}+6z^{2}a^{4}-a^{4}+3z^{7}a^{3}-9z^{5}a^{3}+7z^{3}a^{3}-za^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$
The A2 invariant	Data:K11n103/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n103/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n103 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["K11n103"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-15t+19-15t^{-1}+7t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 65, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-3q^{-1}+6q^{-2}-8q^{-3}+11q^{-4}-11q^{-5}+10q^{-6}-8q^{-7}+5q^{-8}-2q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{3}a^{11}-2za^{11}+z^{6}a^{10}+4z^{4}a^{10}-3z^{2}a^{10}+3z^{7}a^{9}-2z^{5}a^{9}+3z^{3}a^{9}-za^{9}+3z^{8}a^{8}-2z^{6}a^{8}+3z^{2}a^{8}-2a^{8}+z^{9}a^{7}+6z^{7}a^{7}-14z^{5}a^{7}+7z^{3}a^{7}-za^{7}+6z^{8}a^{6}-9z^{6}a^{6}-5z^{4}a^{6}+9z^{2}a^{6}-3a^{6}+z^{9}a^{5}+6z^{7}a^{5}-21z^{5}a^{5}+14z^{3}a^{5}-3za^{5}+3z^{8}a^{4}-5z^{6}a^{4}-4z^{4}a^{4}+6z^{2}a^{4}-a^{4}+3z^{7}a^{3}-9z^{5}a^{3}+7z^{3}a^{3}-za^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n10, K11n144,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n175,}

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["K11n103"];

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]=

{ $-t^{3}+7t^{2}-15t+19-15t^{-1}+7t^{-2}-t^{-3}$, $1-3q^{-1}+6q^{-2}-8q^{-3}+11q^{-4}-11q^{-5}+10q^{-6}-8q^{-7}+5q^{-8}-2q^{-9}$ }

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]=

{K11n10, K11n144,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n175,}

Vassiliev invariants

V2 and V3:

(4, -9)

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 $16$ $-72$ $128$ ${\frac {1112}{3}}$ ${\frac {160}{3}}$ $-1152$ $-2160$ $-352$ $-296$ ${\frac {2048}{3}}$ $2592$ ${\frac {17792}{3}}$ ${\frac {2560}{3}}$ ${\frac {193022}{15}}$ ${\frac {1344}{5}}$ ${\frac {221888}{45}}$ ${\frac {1282}{9}}$ ${\frac {9422}{15}}$

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 $s=$-4 is the signature of K11n103. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n102 K11n104