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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 6q-13/2 + 9q-11/2 - 13q-9/2 + 16q-7/2 - 17q-5/2 + 15q-3/2 - 12q-1/2 + 8q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-26 + 2q-22 + 5q-20 + 3q-16 + q-14 - 4q-12 + q-10 - 2q-8 + 2q-6 + q-4 - 2q-2 + 3 - 3q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 + az - az5 + a3z-1 - 3a3z - 4a3z3 - 2a3z5 - 3a5z-1 - 4a5z - 2a5z3 - a5z5 + 2a7z-1 + 2a7z + a7z3

Kauffman Polynomial:

- a-2z4 + 2a-1z3 - 4a-1z5 - 2z2 + 8z4 - 8z6 + az - 6az3 + 13az5 - 10az7 - a2 - a2z2 + 3a2z4 + 7a2z6 - 8a2z8 + a3z-1 + a3z - 14a3z3 + 23a3z5 - 5a3z7 - 4a3z9 - 3a4 + 15a4z2 - 30a4z4 + 33a4z6 - 11a4z8 - a4z10 + 3a5z-1 - 14a5z + 18a5z3 - 15a5z5 + 17a5z7 - 7a5z9 - 3a6 + 14a6z2 - 30a6z4 + 25a6z6 - 5a6z8 - a6z10 + 2a7z-1 - 10a7z + 16a7z3 - 16a7z5 + 11a7z7 - 3a7z9 - 6a8z4 + 7a8z6 - 2a8z8 + 4a9z - 8a9z3 + 5a9z5 - 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                   5 1   j = 0                 7 3     j = -2               9 6       j = -4             8 6         j = -6           8 9           j = -8         5 8             j = -10       4 8               j = -12     2 5                 j = -14     4                   j = -16 1 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, 295]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 295]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[22, 20, 9, 19], X[18, 7, 19, 8], > X[6, 15, 7, 16], X[16, 5, 17, 6], X[4, 17, 5, 18], X[14, 22, 15, 21], > X[20, 14, 21, 13], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -7, 6, -5, 4, -11}, > {10, -1, 11, -2, 9, -8, 5, -6, 7, -4, 3, -9, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 6 9 13 16 17 15 12 -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 > 8 Sqrt[q] - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 2 5 3 -14 4 -10 2 2 -4 2 2 3 + q + --- + --- + --- + q - --- + q - -- + -- + q - -- - 3 q + 22 20 16 12 8 6 2 q q q q q q q 6 8 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 295]][a, z]
Out[8]=	3 5 7 3 a 3 a 2 a 3 5 7 z 3 3 5 3 -- - ---- + ---- + a z - 3 a z - 4 a z + 2 a z + -- - 4 a z - 2 a z + z z z a 7 3 5 3 5 5 5 > a z - a z - 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 295]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 -a - 3 a - 3 a + -- + ---- + ---- + a z + a z - 14 a z - 10 a z + z z z 3 9 2 2 2 4 2 6 2 2 z 3 3 3 > 4 a z - 2 z - a z + 15 a z + 14 a z + ---- - 6 a z - 14 a z + a 4 5 3 7 3 9 3 4 z 2 4 4 4 6 4 > 18 a z + 16 a z - 8 a z + 8 z - -- + 3 a z - 30 a z - 30 a z - 2 a 5 8 4 4 z 5 3 5 5 5 7 5 9 5 > 6 a z - ---- + 13 a z + 23 a z - 15 a z - 16 a z + 5 a z - a 6 2 6 4 6 6 6 8 6 7 3 7 > 8 z + 7 a z + 33 a z + 25 a z + 7 a z - 10 a z - 5 a z + 5 7 7 7 9 7 2 8 4 8 6 8 8 8 > 17 a z + 11 a z - a z - 8 a z - 11 a z - 5 a z - 2 a z - 3 9 5 9 7 9 4 10 6 10 > 4 a z - 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 2 4 2 5 4 7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 8 16 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 8 5 8 8 9 8 6 9 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 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 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a295

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