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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X4,22,5,21 X14,5,15,6 X20,13,21,14 X6,19,7,20 X16,8,17,7 X18,16,19,15 X8,18,1,17

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 8q-5/2 - 14q-3/2 + 19q-1/2 - 23q1/2 + 23q3/2 - 21q5/2 + 15q7/2 - 10q9/2 + 5q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 - 2q-8 + 5q-6 - q-4 - q-2 + 4 - 3q2 + 5q4 - q6 + 2q8 + 3q10 - 4q12 + 3q14 - q16 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-5z3 + 3a-3z + 4a-3z3 + 2a-3z5 - a-1z-1 - 7a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + az-1 + 5az + 5az3 + 2az5 - a3z - a3z3

Kauffman Polynomial:

- a-7z5 + 5a-6z4 - 5a-6z6 - 6a-5z3 + 15a-5z5 - 10a-5z7 - 3a-4z2 + 4a-4z4 + 9a-4z6 - 10a-4z8 + 6a-3z - 30a-3z3 + 43a-3z5 - 14a-3z7 - 5a-3z9 - 19a-2z4 + 41a-2z6 - 21a-2z8 - a-2z10 - a-1z-1 + 14a-1z - 46a-1z3 + 47a-1z5 - 5a-1z7 - 9a-1z9 + 1 + 4z2 - 26z4 + 40z6 - 17z8 - z10 - az-1 + 10az - 30az3 + 30az5 - 5az7 - 4az9 - 6a2z4 + 12a2z6 - 6a2z8 + 2a3z - 8a3z3 + 10a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     4   j = 10                   6 1   j = 8                 9 4     j = 6               12 6       j = 4             11 9         j = 2           12 12           j = 0         9 13             j = -2       5 10               j = -4     3 9                 j = -6   1 5                   j = -8   3                     j = -10 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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 248]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 248]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[4, 22, 5, 21], X[14, 5, 15, 6], X[20, 13, 21, 14], X[6, 19, 7, 20], > X[16, 8, 17, 7], X[18, 16, 19, 15], X[8, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -8, 9, -11}, > {4, -1, 3, -2, 7, -6, 10, -9, 11, -10, 8, -7, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 8 14 19 3/2 5/2 q - ---- + ---- - ---- + ------- - 23 Sqrt[q] + 23 q - 21 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 15 q - 10 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 5 -4 -2 2 4 6 8 4 - q + q + q - -- + -- - q - q - 3 q + 5 q - q + 2 q + 8 6 q q 10 12 14 16 18 20 > 3 q - 4 q + 3 q - q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 248]][a, z]
Out[8]=	3 3 3 1 a 3 z 7 z 3 z 4 z 9 z 3 3 3 -(---) + - + --- - --- + 5 a z - a z - -- + ---- - ---- + 5 a z - a z + a z z 3 a 5 3 a a a a 5 5 7 2 z 4 z 5 z > ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 248]][a, z]
Out[9]=	2 3 1 a 6 z 14 z 3 2 3 z 4 2 6 z 1 - --- - - + --- + ---- + 10 a z + 2 a z + 4 z - ---- - a z - ---- - a z z 3 a 4 5 a a a 3 3 4 4 4 30 z 46 z 3 3 3 4 5 z 4 z 19 z 2 4 > ----- - ----- - 30 a z - 8 a z - 26 z + ---- + ---- - ----- - 6 a z + 3 a 6 4 2 a a a a 5 5 5 5 6 4 4 z 15 z 43 z 47 z 5 3 5 6 5 z > 2 a z - -- + ----- + ----- + ----- + 30 a z + 10 a z + 40 z - ---- + 7 5 3 a 6 a a a a 6 6 7 7 7 9 z 41 z 2 6 4 6 10 z 14 z 5 z 7 3 7 > ---- + ----- + 12 a z - a z - ----- - ----- - ---- - 5 a z - 4 a z - 4 2 5 3 a a a a a 8 8 9 9 10 8 10 z 21 z 2 8 5 z 9 z 9 10 z > 17 z - ----- - ----- - 6 a z - ---- - ---- - 4 a z - z - --- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 9 5 9 13 + 12 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 10 2 4 4 2 6 2 6 3 8 3 > ---- + 12 q t + 11 q t + 9 q t + 12 q t + 6 q t + 9 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a248

L11a247 L11a249