K11n3

K11n2

K11n4

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n3 at Knotilus!

Knot K11n3.

A graph, knot K11n3.

A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X7,15,8,14 X2,9,3,10 X11,16,12,17 X13,20,14,21 X15,7,16,6 X17,22,18,1 X19,12,20,13 X21,18,22,19
Gauss code	1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 10, -7, 4, -8, 6, -9, 11, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -20 -6 -22 -12 -18

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:K11n3/ThurstonBennequinNumber Hyperbolic Volume 11.5634 A-Polynomial See Data:K11n3/A-polynomial

[edit Notes for K11n3'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 K11n3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+11t-15+11t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, -2 }
Jones polynomial	$q^{3}-2q^{2}+4q-6+7q^{-1}-7q^{-2}+7q^{-3}-5q^{-4}+3q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{6}+3z^{2}a^{4}+3a^{4}-2z^{4}a^{2}-4z^{2}a^{2}-2a^{2}-z^{4}-z^{2}+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+4a^{2}z^{8}+2z^{8}+a^{5}z^{7}-a^{3}z^{7}+2z^{7}a^{-1}-7a^{4}z^{6}-13a^{2}z^{6}+z^{6}a^{-2}-5z^{6}-a^{5}z^{5}-a^{3}z^{5}-7az^{5}-7z^{5}a^{-1}+3a^{6}z^{4}+15a^{4}z^{4}+16a^{2}z^{4}-4z^{4}a^{-2}+a^{7}z^{3}+4a^{5}z^{3}+4a^{3}z^{3}+7az^{3}+6z^{3}a^{-1}-3a^{6}z^{2}-11a^{4}z^{2}-10a^{2}z^{2}+4z^{2}a^{-2}+2z^{2}-a^{7}z-2a^{5}z-2a^{3}z-3az-2za^{-1}+a^{6}+3a^{4}+2a^{2}-a^{-2}$
The A2 invariant	Data:K11n3/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n3/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n3 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["K11n3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+11t-15+11t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 43, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+4a^{2}z^{8}+2z^{8}+a^{5}z^{7}-a^{3}z^{7}+2z^{7}a^{-1}-7a^{4}z^{6}-13a^{2}z^{6}+z^{6}a^{-2}-5z^{6}-a^{5}z^{5}-a^{3}z^{5}-7az^{5}-7z^{5}a^{-1}+3a^{6}z^{4}+15a^{4}z^{4}+16a^{2}z^{4}-4z^{4}a^{-2}+a^{7}z^{3}+4a^{5}z^{3}+4a^{3}z^{3}+7az^{3}+6z^{3}a^{-1}-3a^{6}z^{2}-11a^{4}z^{2}-10a^{2}z^{2}+4z^{2}a^{-2}+2z^{2}-a^{7}z-2a^{5}z-2a^{3}z-3az-2za^{-1}+a^{6}+3a^{4}+2a^{2}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_7, K11a59,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_22,}

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["K11n3"];

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]=

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

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]=

{10_7, K11a59,}

In[6]:=

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

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

Out[6]=

{9_22,}

Vassiliev invariants

V2 and V3:

(-1, -1)

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 $-4$ $-8$ $8$ ${\frac {178}{3}}$ ${\frac {86}{3}}$ $32$ $-{\frac {80}{3}}$ $-{\frac {32}{3}}$ $-8$ $-{\frac {32}{3}}$ $32$ $-{\frac {712}{3}}$ $-{\frac {344}{3}}$ $-{\frac {4831}{30}}$ ${\frac {1462}{15}}$ $-{\frac {12902}{45}}$ ${\frac {1087}{18}}$ $-{\frac {1951}{30}}$

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 K11n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n2 K11n4