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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X18,5,19,6 X20,16,21,15 X16,20,17,19 X14,22,15,21 X2738 X4,11,5,12 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 6q-9/2 + 9q-7/2 - 14q-5/2 + 16q-3/2 - 16q-1/2 + 15q1/2 - 11q3/2 + 7q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-16 + 4q-14 - 2q-12 + 2q-10 + 2q-8 - 2q-6 + 2q-4 - 3q-2 + 1 - q2 - 2q4 + 4q6 - 2q8 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - 2a-1z - a-1z3 - a-1z5 + az + az3 - az5 + a3z + 3a3z3 - a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

2a-4z4 - a-4z6 - 6a-3z3 + 11a-3z5 - 4a-3z7 + 4a-2z2 - 13a-2z4 + 17a-2z6 - 6a-2z8 + a-1z - 12a-1z3 + 11a-1z5 + 4a-1z7 - 4a-1z9 + 17z2 - 41z4 + 36z6 - 10z8 - z10 - 3az + 5az3 - 11az5 + 14az7 - 7az9 + 10a2z2 - 24a2z4 + 20a2z6 - 7a2z8 - a2z10 - 3a3z + 9a3z3 - 7a3z5 + 3a3z7 - 3a3z9 - 3a4z2 + 5a4z4 - 3a4z8 + a5z-1 - 2a5z + a5z3 + 3a5z5 - 3a5z7 - a6 + 3a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     3   j = 6                   4 1   j = 4                 7 3     j = 2               8 4       j = 0             8 7         j = -2           9 9           j = -4         5 7             j = -6       4 9               j = -8     2 5                 j = -10     4                   j = -12 1 2                     j = -14 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, 241]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 241]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 10, 7, 9], X[10, 14, 11, 13], > X[18, 5, 19, 6], X[20, 16, 21, 15], X[16, 20, 17, 19], X[14, 22, 15, 21], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 5, -11}, > {9, -1, 3, -4, 10, -2, 4, -8, 6, -7, 11, -5, 7, -6, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 6 9 14 16 16 -q + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 11 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 4 2 2 2 2 2 3 2 4 6 1 + q + --- + --- + --- - --- + --- + -- - -- + -- - -- - q - 2 q + 4 q - 20 16 14 12 10 8 6 4 2 q q q q q q q q q 8 12 14 > 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 241]][a, z]
Out[8]=	5 7 3 3 5 a a 2 z 3 5 z z 3 3 3 z 5 -(--) + -- - --- + a z + a z - 3 a z + -- - -- + a z + 3 a z - -- - a z z z a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 241]][a, z]
Out[9]=	5 7 2 6 a a z 3 5 7 2 4 z -a + -- + -- + - - 3 a z - 3 a z - 2 a z - 3 a z + 17 z + ---- + z z a 2 a 3 3 2 2 4 2 6 z 12 z 3 3 3 5 3 7 3 > 10 a z - 3 a z - ---- - ----- + 5 a z + 9 a z + a z + 3 a z - 3 a a 4 4 5 5 4 2 z 13 z 2 4 4 4 6 4 11 z 11 z > 41 z + ---- - ----- - 24 a z + 5 a z + 3 a z + ----- + ----- - 4 2 3 a a a a 6 6 5 3 5 5 5 7 5 6 z 17 z 2 6 > 11 a z - 7 a z + 3 a z - a z + 36 z - -- + ----- + 20 a z - 4 2 a a 7 7 8 6 6 4 z 4 z 7 3 7 5 7 8 6 z > 2 a z - ---- + ---- + 14 a z + 3 a z - 3 a z - 10 z - ---- - 3 a 2 a a 9 2 8 4 8 4 z 9 3 9 10 2 10 > 7 a z - 3 a z - ---- - 7 a z - 3 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 1 2 4 2 5 4 9 8 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 5 7 9 2 2 2 4 2 4 3 > ----- + ---- + ---- + 7 t + 8 q t + 4 q t + 7 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a241

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