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	$-3 t^2+11 t-15+11 t^{-1} -3 t^{-2}$
Conway polynomial	$-3 z^4-z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, -2 }
Jones polynomial	$q^3-2 q^2+4 q-6+7 q^{-1} -7 q^{-2} +7 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^6+3 z^2 a^4+3 a^4-2 z^4 a^2-4 z^2 a^2-2 a^2-z^4-z^2+z^2 a^{-2} + a^{-2}$
Kauffman polynomial (db, data sources)	$a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+a^5 z^7-a^3 z^7+2 z^7 a^{-1} -7 a^4 z^6-13 a^2 z^6+z^6 a^{-2} -5 z^6-a^5 z^5-a^3 z^5-7 a z^5-7 z^5 a^{-1} +3 a^6 z^4+15 a^4 z^4+16 a^2 z^4-4 z^4 a^{-2} +a^7 z^3+4 a^5 z^3+4 a^3 z^3+7 a z^3+6 z^3 a^{-1} -3 a^6 z^2-11 a^4 z^2-10 a^2 z^2+4 z^2 a^{-2} +2 z^2-a^7 z-2 a^5 z-2 a^3 z-3 a z-2 z a^{-1} +a^6+3 a^4+2 a^2- a^{-2}$
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]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"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]=

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

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 $t^rq^j$ 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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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