K11n112

K11n111

K11n113

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n112 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,6,15,5 X18,8,19,7 X2,10,3,9 X20,11,21,12 X8,14,9,13 X15,22,16,1 X6,18,7,17 X12,19,13,20 X21,16,22,17
Gauss code	1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -10, 7, -3, -8, 11, 9, -4, 10, -6, -11, 8
Dowker-Thistlethwaite code	4 10 14 18 2 20 8 -22 6 12 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n112/ThurstonBennequinNumber Hyperbolic Volume 13.3686 A-Polynomial See Data:K11n112/A-polynomial

[edit Notes for K11n112's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11n112's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}$
Conway polynomial	$z^6+z^4+2 z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, 2 }
Jones polynomial	$-q^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^6 a^{-4} -2 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -4 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +z^2-2 a^{-2} +4 a^{-4} -2 a^{-6} +1$
Kauffman polynomial (db, data sources)	$z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +5 z^8 a^{-4} +4 z^8 a^{-6} -z^7 a^{-3} +4 z^7 a^{-5} +5 z^7 a^{-7} -2 z^6 a^{-2} -15 z^6 a^{-4} -10 z^6 a^{-6} +3 z^6 a^{-8} +3 z^5 a^{-1} +2 z^5 a^{-3} -15 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} +8 z^4 a^{-2} +24 z^4 a^{-4} +11 z^4 a^{-6} -6 z^4 a^{-8} +z^4-4 z^3 a^{-1} +2 z^3 a^{-3} +18 z^3 a^{-5} +10 z^3 a^{-7} -2 z^3 a^{-9} -9 z^2 a^{-2} -15 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -2 z^2-2 z a^{-3} -6 z a^{-5} -4 z a^{-7} +2 a^{-2} +4 a^{-4} +2 a^{-6} +1$
The A2 invariant	Data:K11n112/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n112/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n112 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["K11n112"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^6+z^4+2 z^2+1$

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]=

{ 55, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^6 a^{-4} -2 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -4 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +z^2-2 a^{-2} +4 a^{-4} -2 a^{-6} +1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +5 z^8 a^{-4} +4 z^8 a^{-6} -z^7 a^{-3} +4 z^7 a^{-5} +5 z^7 a^{-7} -2 z^6 a^{-2} -15 z^6 a^{-4} -10 z^6 a^{-6} +3 z^6 a^{-8} +3 z^5 a^{-1} +2 z^5 a^{-3} -15 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} +8 z^4 a^{-2} +24 z^4 a^{-4} +11 z^4 a^{-6} -6 z^4 a^{-8} +z^4-4 z^3 a^{-1} +2 z^3 a^{-3} +18 z^3 a^{-5} +10 z^3 a^{-7} -2 z^3 a^{-9} -9 z^2 a^{-2} -15 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -2 z^2-2 z a^{-3} -6 z a^{-5} -4 z a^{-7} +2 a^{-2} +4 a^{-4} +2 a^{-6} +1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_31, K11n11, K11n22, K11n127,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_39, K11n11,}

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["K11n112"];

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]=

{ $t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}$, $-q^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1}$ }

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]=

{9_31, K11n11, K11n22, K11n127,}

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_39, K11n11,}

Vassiliev invariants

V2 and V3:

(2, 4)

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 $8$ $32$ $32$ $\frac{316}{3}$ $\frac{20}{3}$ $256$ $\frac{1472}{3}$ $\frac{32}{3}$ $128$ $\frac{256}{3}$ $512$ $\frac{2528}{3}$ $\frac{160}{3}$ $\frac{33391}{15}$ $-\frac{5324}{15}$ $\frac{52204}{45}$ $\frac{881}{9}$ $\frac{2191}{15}$

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 $t^rq^j$ 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 K11n112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

K11n111 K11n113