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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X22,10,5,9 X18,7,19,8 X8,17,9,18 X12,19,13,20 X20,11,21,12 X10,16,11,15 X16,22,17,21 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 3q-15/2 - 8q-13/2 + 14q-11/2 - 19q-9/2 + 22q-7/2 - 23q-5/2 + 20q-3/2 - 16q-1/2 + 9q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 - q-24 + 3q-20 - 5q-18 + q-16 + q-14 - 3q-12 + 5q-10 - q-8 + 5q-6 + q-4 - 2q-2 + 5 - 4q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 + az + az3 - az5 - 2a3z-1 - 8a3z - 7a3z3 - 3a3z5 + 4a5z-1 + 9a5z + 6a5z3 - 3a7z-1 - 4a7z + a9z-1

Kauffman Polynomial:

- a-2z4 + a-1z3 - 4a-1z5 - z2 + 6z4 - 9z6 + 3az - 14az3 + 22az5 - 15az7 - a2 + 3a2z2 - 14a2z4 + 27a2z6 - 16a2z8 - 2a3z-1 + 16a3z - 42a3z3 + 41a3z5 - a3z7 - 9a3z9 - 2a4 + 15a4z2 - 53a4z4 + 68a4z6 - 22a4z8 - 2a4z10 - 4a5z-1 + 24a5z - 44a5z3 + 19a5z5 + 22a5z7 - 13a5z9 - 3a6 + 17a6z2 - 44a6z4 + 42a6z6 - 9a6z8 - 2a6z10 - 3a7z-1 + 15a7z - 23a7z3 + 8a7z5 + 7a7z7 - 4a7z9 - a8 + 6a8z2 - 12a8z4 + 10a8z6 - 3a8z8 - a9z-1 + 4a9z - 6a9z3 + 4a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                       1 j = 4                     3   j = 2                   6 1   j = 0                 10 3     j = -2               11 7       j = -4             12 9         j = -6           10 11           j = -8         9 12             j = -10       6 11               j = -12     2 8                 j = -14   1 6                   j = -16   2                     j = -18 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, 127]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 127]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[22, 10, 5, 9], X[18, 7, 19, 8], > X[8, 17, 9, 18], X[12, 19, 13, 20], X[20, 11, 21, 12], X[10, 16, 11, 15], > X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 3, -8, 7, -6, 11, -2, 8, -9, 5, -4, > 6, -7, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 8 14 19 22 23 20 16 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 9 Sqrt[q] - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -24 3 5 -16 -14 3 5 -8 5 -4 5 + q + --- - q + --- - --- + q + q - --- + --- - q + -- + q - 26 20 18 12 10 6 q q q q q q 2 2 6 8 > -- - 4 q + 2 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 127]][a, z]
Out[8]=	3 5 7 9 3 -2 a 4 a 3 a a 3 5 7 z 3 ----- + ---- - ---- + -- + a z - 8 a z + 9 a z - 4 a z + -- + a z - z z z z a 3 3 5 3 5 3 5 > 7 a z + 6 a z - a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 127]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 2 a 4 a 3 a a 3 5 -a - 2 a - 3 a - a - ---- - ---- - ---- - -- + 3 a z + 16 a z + 24 a z + z z z z 3 7 9 2 2 2 4 2 6 2 8 2 z > 15 a z + 4 a z - z + 3 a z + 15 a z + 17 a z + 6 a z + -- - a 4 3 3 3 5 3 7 3 9 3 4 z 2 4 > 14 a z - 42 a z - 44 a z - 23 a z - 6 a z + 6 z - -- - 14 a z - 2 a 5 4 4 6 4 8 4 4 z 5 3 5 5 5 > 53 a z - 44 a z - 12 a z - ---- + 22 a z + 41 a z + 19 a z + a 7 5 9 5 6 2 6 4 6 6 6 8 6 > 8 a z + 4 a z - 9 z + 27 a z + 68 a z + 42 a z + 10 a z - 7 3 7 5 7 7 7 9 7 2 8 4 8 > 15 a z - a z + 22 a z + 7 a z - a z - 16 a z - 22 a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 9 a z - 3 a z - 9 a z - 13 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 2 1 6 2 8 6 10 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 11 9 12 10 11 12 9 11 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t 2 2 2 4 2 6 3 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a127

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