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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 4q-9/2 + 6q-7/2 - 8q-5/2 + 9q-3/2 - 10q-1/2 + 7q1/2 - 6q3/2 + 4q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 2q-14 + q-10 + q-8 - q-6 + 3q-4 + 4 + q2 - 2q8 - q12

HOMFLY-PT Polynomial:

a-1z-1 + 9a-1z + 12a-1z3 + 6a-1z5 + a-1z7 - 3az-1 - 20az - 33az3 - 24az5 - 8az7 - az9 + 2a3z-1 + 9a3z + 12a3z3 + 6a3z5 + a3z7

Kauffman Polynomial:

- 3a-4z2 + 4a-4z4 - a-4z6 - 4a-3z3 + 7a-3z5 - 2a-3z7 - a-2 + 8a-2z2 - 14a-2z4 + 12a-2z6 - 3a-2z8 + a-1z-1 - 11a-1z + 26a-1z3 - 26a-1z5 + 14a-1z7 - 3a-1z9 - 3 + 20z2 - 29z4 + 13z6 - z10 + 3az-1 - 23az + 48az3 - 52az5 + 25az7 - 5az9 - 3a2 + 14a2z2 - 20a2z4 + 6a2z6 + a2z8 - a2z10 + 2a3z-1 - 10a3z + 16a3z3 - 15a3z5 + 7a3z7 - 2a3z9 + 3a4z2 - 4a4z4 + 4a4z6 - 2a4z8 + a5z3 + 3a5z5 - 2a5z7 - 2a6z2 + 5a6z4 - 2a6z6 - 2a7z + 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                     1   j = 6                   3 1   j = 4                 3 1     j = 2               4 3       j = 0             6 3         j = -2           4 5           j = -4         4 5             j = -6       2 4               j = -8     2 4                 j = -10   1 3                   j = -12   1                     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, 274]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 274]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[6, 9, 7, 10], X[18, 8, 19, 7], X[20, 16, 21, 15], X[22, 18, 9, 17], > X[16, 22, 17, 21], X[8, 20, 1, 19], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -5, 6, -10}, > {5, -1, 2, -3, 11, -4, 7, -9, 8, -6, 10, -7, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 4 6 8 9 10 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 6 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 9/2 > 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -10 -8 -6 3 2 8 12 4 + q + --- + q + q - q + -- + q - 2 q - q 14 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 274]][a, z]
Out[8]=	3 3 5 1 3 a 2 a 9 z 3 12 z 3 3 3 6 z --- - --- + ---- + --- - 20 a z + 9 a z + ----- - 33 a z + 12 a z + ---- - a z z z a a a 7 5 3 5 z 7 3 7 9 > 24 a z + 6 a z + -- - 8 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 274]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 11 z 3 7 2 -3 - a - 3 a + --- + --- + ---- - ---- - 23 a z - 10 a z - 2 a z + 20 z - a z z z a 2 2 3 3 3 z 8 z 2 2 4 2 6 2 4 z 26 z 3 > ---- + ---- + 14 a z + 3 a z - 2 a z - ---- + ----- + 48 a z + 4 2 3 a a a a 4 4 3 3 5 3 7 3 4 4 z 14 z 2 4 4 4 > 16 a z + a z + 3 a z - 29 z + ---- - ----- - 20 a z - 4 a z + 4 2 a a 5 5 6 4 7 z 26 z 5 3 5 5 5 7 5 6 > 5 a z + ---- - ----- - 52 a z - 15 a z + 3 a z - a z + 13 z - 3 a a 6 6 7 7 z 12 z 2 6 4 6 6 6 2 z 14 z 7 > -- + ----- + 6 a z + 4 a z - 2 a z - ---- + ----- + 25 a z + 4 2 3 a a a a 8 9 3 7 5 7 3 z 2 8 4 8 3 z 9 3 9 > 7 a z - 2 a z - ---- + a z - 2 a z - ---- - 5 a z - 2 a z - 2 a a 10 2 10 > z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 1 3 2 4 2 4 6 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 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 4 5 4 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 3 t + 4 q t + 3 q t + 3 q t + q t + 3 q t + 4 2 4 2 q t q t q t 6 4 8 4 10 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a274

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