K11a307

K11a306

K11a308

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a307 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X12,3,13,4 X14,5,15,6 X16,8,17,7 X20,9,21,10 X18,11,19,12 X4,13,5,14 X2,15,3,16 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code	1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -6, 10, -5, 11, -9
Dowker-Thistlethwaite code	6 12 14 16 20 18 4 2 22 10 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a307/ThurstonBennequinNumber Hyperbolic Volume 11.9838 A-Polynomial See Data:K11a307/A-polynomial

[edit Notes for K11a307'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 K11a307's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2 t^3-9 t^2+19 t-23+19 t^{-1} -9 t^{-2} +2 t^{-3}$
Conway polynomial	$2 z^6+3 z^4+z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 83, -2 }
Jones polynomial	$-q^2+3 q-5+9 q^{-1} -11 q^{-2} +13 q^{-3} -13 q^{-4} +11 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^2 a^8+a^8-2 z^4 a^6-5 z^2 a^6-3 a^6+z^6 a^4+3 z^4 a^4+4 z^2 a^4+2 a^4+z^6 a^2+3 z^4 a^2+3 z^2 a^2+a^2-z^4-2 z^2$
Kauffman polynomial (db, data sources)	$z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+7 z^3 a^9-z a^9+4 z^8 a^8-13 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+3 z^9 a^7-7 z^7 a^7+2 z^5 a^7+2 z^3 a^7-z a^7+z^{10} a^6+4 z^8 a^6-22 z^6 a^6+33 z^4 a^6-18 z^2 a^6+3 a^6+6 z^9 a^5-20 z^7 a^5+29 z^5 a^5-15 z^3 a^5+2 z a^5+z^{10} a^4+4 z^8 a^4-19 z^6 a^4+32 z^4 a^4-17 z^2 a^4+2 a^4+3 z^9 a^3-6 z^7 a^3+8 z^5 a^3-5 z^3 a^3+2 z a^3+4 z^8 a^2-8 z^6 a^2+6 z^4 a^2-z^2 a^2-a^2+4 z^7 a-8 z^5 a+3 z^3 a+3 z^6-7 z^4+3 z^2+z^5 a^{-1} -2 z^3 a^{-1}$
The A2 invariant	Data:K11a307/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a307/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a307 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["K11a307"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2 t^3-9 t^2+19 t-23+19 t^{-1} -9 t^{-2} +2 t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2 z^6+3 z^4+z^2+1$

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]=

{ 83, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^2 a^8+a^8-2 z^4 a^6-5 z^2 a^6-3 a^6+z^6 a^4+3 z^4 a^4+4 z^2 a^4+2 a^4+z^6 a^2+3 z^4 a^2+3 z^2 a^2+a^2-z^4-2 z^2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+7 z^3 a^9-z a^9+4 z^8 a^8-13 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+3 z^9 a^7-7 z^7 a^7+2 z^5 a^7+2 z^3 a^7-z a^7+z^{10} a^6+4 z^8 a^6-22 z^6 a^6+33 z^4 a^6-18 z^2 a^6+3 a^6+6 z^9 a^5-20 z^7 a^5+29 z^5 a^5-15 z^3 a^5+2 z a^5+z^{10} a^4+4 z^8 a^4-19 z^6 a^4+32 z^4 a^4-17 z^2 a^4+2 a^4+3 z^9 a^3-6 z^7 a^3+8 z^5 a^3-5 z^3 a^3+2 z a^3+4 z^8 a^2-8 z^6 a^2+6 z^4 a^2-z^2 a^2-a^2+4 z^7 a-8 z^5 a+3 z^3 a+3 z^6-7 z^4+3 z^2+z^5 a^{-1} -2 z^3 a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_83, K11a323,}

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["K11a307"];

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]=

{ $2 t^3-9 t^2+19 t-23+19 t^{-1} -9 t^{-2} +2 t^{-3}$, $-q^2+3 q-5+9 q^{-1} -11 q^{-2} +13 q^{-3} -13 q^{-4} +11 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9}$ }

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]=

{10_83, K11a323,}

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:

(1, 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 $4$ $0$ $8$ $-\frac{130}{3}$ $-\frac{62}{3}$ $0$ $224$ $-32$ $160$ $\frac{32}{3}$ $0$ $-\frac{520}{3}$ $-\frac{248}{3}$ $-\frac{23249}{30}$ $\frac{10178}{15}$ $-\frac{45538}{45}$ $-\frac{2863}{18}$ $-\frac{5009}{30}$

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 K11a307. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     2   2 1                   3 1   -2 -1                 6 2     4 -3               6 4       -2 -5             7 5         2 -7           6 6           0 -9         5 7             -2 -11       3 6               3 -13     2 5                 -3 -15   1 3                   2 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

K11a306 K11a308