© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n129 K11n131

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,18,6,19 X7,14,8,15 X9,17,10,16 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,13,18,12 X19,6,20,7 X21,9,22,8

Gauss Code:

{1, -6, 2, -1, -3, 10, -4, 11, -5, -2, 6, 9, -7, 4, -8, 5, -9, 3, -10, 7, -11, 8}

DT (Dowker-Thistlethwaite) Code:

4 10 -18 -14 -16 2 -20 -22 -12 -6 -8

Alexander Polynomial:

- t-3 + 5t-2 - 12t-1 + 17 - 12t + 5t2 - t3

Conway Polynomial:

1 - z2 - z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{930, ...}

Determinant and Signature:

{53, 0}

Jones Polynomial:

q-6 - 3q-5 + 5q-4 - 7q-3 + 9q-2 - 9q-1 + 8 - 6q + 4q2 - q3

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + q-14 - 2q-10 + 2q-8 - q-6 + q-4 - 1 + 2q2 - q4 + 2q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + 2 + 4z2 + 2z4 - 2a2 - 6a2z2 - 4a2z4 - a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

a-3z3 - 3a-2z2 + 4a-2z4 - 2a-1z3 + 2a-1z5 + a-1z7 + 2 - 12z2 + 19z4 - 10z6 + 3z8 - 4az3 + 7az5 - 5az7 + 2az9 + 2a2 - 15a2z2 + 31a2z4 - 25a2z6 + 7a2z8 - a3z + 6a3z3 - 5a3z5 - 3a3z7 + 2a3z9 + a4 - 5a4z2 + 13a4z4 - 14a4z6 + 4a4z8 - a5z + 7a5z3 - 10a5z5 + 3a5z7 + a6z2 - 3a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 1}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 11130. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   1 j = 5                 3   j = 3               3 1   j = 1             5 3     j = -1           5 4       j = -3         4 4         j = -5       3 5           j = -7     2 4             j = -9   1 3               j = -11   2                 j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 130]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 130]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 18, 6, 19], X[7, 14, 8, 15], > X[9, 17, 10, 16], X[2, 11, 3, 12], X[13, 21, 14, 20], X[15, 22, 16, 1], > X[17, 13, 18, 12], X[19, 6, 20, 7], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 130]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 10, -4, 11, -5, -2, 6, 9, -7, 4, -8, 5, -9, 3, -10, > 7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 130]]
Out[5]=	DTCode[4, 10, -18, -14, -16, 2, -20, -22, -12, -6, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 130]][t]
Out[6]=	-3 5 12 2 3 17 - t + -- - -- - 12 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 130]][z]
Out[7]=	2 4 6 1 - z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 30], Knot[11, NonAlternating, 130]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 130]], KnotSignature[Knot[11, NonAlternating, 130]]}
Out[9]=	{53, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 130]][q]
Out[10]=	-6 3 5 7 9 9 2 3 8 + q - -- + -- - -- + -- - - - 6 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 130]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 130]][q]
Out[12]=	-18 -16 -14 2 2 -6 -4 2 4 6 8 10 -1 + q - q + q - --- + -- - q + q + 2 q - q + 2 q + q - q 10 8 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 130]][a, z]
Out[13]=	2 2 4 2 z 2 2 4 2 4 2 4 4 4 2 6 2 - 2 a + a + 4 z - -- - 6 a z + 2 a z + 2 z - 4 a z + a z - a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 130]][a, z]
Out[14]=	2 3 2 4 3 5 2 3 z 2 2 4 2 6 2 z 2 + 2 a + a - a z - a z - 12 z - ---- - 15 a z - 5 a z + a z + -- - 2 3 a a 3 4 2 z 3 3 3 5 3 4 4 z 2 4 4 4 > ---- - 4 a z + 6 a z + 7 a z + 19 z + ---- + 31 a z + 13 a z - a 2 a 5 6 4 2 z 5 3 5 5 5 6 2 6 > 3 a z + ---- + 7 a z - 5 a z - 10 a z - 10 z - 25 a z - a 7 4 6 6 6 z 7 3 7 5 7 8 2 8 > 14 a z + a z + -- - 5 a z - 3 a z + 3 a z + 3 z + 7 a z + a 4 8 9 3 9 > 4 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 130]], Vassiliev[3][Knot[11, NonAlternating, 130]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 130]][q, t]
Out[16]=	4 1 2 1 3 2 4 3 5 - + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 4 4 5 3 3 2 5 2 7 3 > ----- + ---- + --- + 3 q t + 3 q t + q t + 3 q t + q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n130

K11n129 K11n131