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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 8q-13/2 + 11q-11/2 - 18q-9/2 + 19q-7/2 - 22q-5/2 + 18q-3/2 - 15q-1/2 + 9q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-28 + 3q-26 + 3q-24 + 7q-22 + 13q-20 + 8q-18 + 13q-16 + 11q-14 + 4q-12 + 9q-10 + 2q-8 + 6q-6 + 2q-4 - 2q-2 + 4 - 4q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 + 2az + az3 - az5 - a3z-3 - 4a3z-1 - 10a3z - 8a3z3 - 3a3z5 + 3a5z-3 + 9a5z-1 + 12a5z + 6a5z3 - 3a7z-3 - 6a7z-1 - 4a7z + a9z-3 + a9z-1

Kauffman Polynomial:

- a-2z4 + a-1z3 - 4a-1z5 - 2z2 + 7z4 - 9z6 + 6az - 15az3 + 21az5 - 14az7 - 2a2z2 - 2a2z4 + 17a2z6 - 13a2z8 + a3z-3 - 5a3z-1 + 20a3z - 53a3z3 + 57a3z5 - 12a3z7 - 6a3z9 - 3a4z-2 + 10a4 - 41a4z4 + 61a4z6 - 22a4z8 - a4z10 + 3a5z-3 - 12a5z-1 + 27a5z - 49a5z3 + 34a5z5 + 8a5z7 - 9a5z9 - 6a6z-2 + 19a6 - 10a6z2 - 31a6z4 + 40a6z6 - 11a6z8 - a6z10 + 3a7z-3 - 12a7z-1 + 23a7z - 22a7z3 + 7a7z5 + 5a7z7 - 3a7z9 - 3a8z-2 + 10a8 - 10a8z2 + 5a8z6 - 2a8z8 + a9z-3 - 5a9z-1 + 10a9z - 10a9z3 + 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                   6 1   j = 0                 9 3     j = -2               10 7       j = -4             12 8         j = -6           11 14           j = -8         7 8             j = -10       6 13               j = -12     2 5                 j = -14     6                   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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 546]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 546]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 16, 11, 15], X[20, 9, 21, 10], > X[22, 13, 19, 14], X[14, 21, 15, 22], X[10, 19, 5, 20], X[8, 18, 9, 17], > X[16, 8, 17, 7], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, 9, -8, 4, -7}, > {11, -2, 5, -6, 3, -9, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 8 11 18 19 22 18 15 -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 3 3 7 13 8 13 11 4 9 2 6 2 4 + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- + -- + -- - 26 24 22 20 18 16 14 12 10 8 6 4 q q q q q q q q q q q q 2 2 6 8 > -- - 4 q + 2 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 546]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a 4 a 9 a 6 a a 3 -(--) + ---- - ---- + -- - ---- + ---- - ---- + -- + 2 a z - 10 a z + 3 3 3 3 z z z z z z z z 3 5 7 z 3 3 3 5 3 5 3 5 > 12 a z - 4 a z + -- + a z - 8 a z + 6 a z - a z - 3 a z a
In[9]:=	Kauffman[Link[11, Alternating, 546]][a, z]
Out[9]=	3 5 7 9 4 6 8 3 4 6 8 a 3 a 3 a a 3 a 6 a 3 a 5 a 10 a + 19 a + 10 a + -- + ---- + ---- + -- - ---- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z z 5 7 9 12 a 12 a 5 a 3 5 7 9 > ----- - ----- - ---- + 6 a z + 20 a z + 27 a z + 23 a z + 10 a z - z z z 3 2 2 2 6 2 8 2 z 3 3 3 5 3 > 2 z - 2 a z - 10 a z - 10 a z + -- - 15 a z - 53 a z - 49 a z - a 4 5 7 3 9 3 4 z 2 4 4 4 6 4 4 z > 22 a z - 10 a z + 7 z - -- - 2 a z - 41 a z - 31 a z - ---- + 2 a a 5 3 5 5 5 7 5 9 5 6 2 6 > 21 a z + 57 a z + 34 a z + 7 a z + 5 a z - 9 z + 17 a z + 4 6 6 6 8 6 7 3 7 5 7 7 7 > 61 a z + 40 a z + 5 a z - 14 a z - 12 a z + 8 a z + 5 a z - 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > a z - 13 a z - 22 a z - 11 a z - 2 a z - 6 a z - 9 a z - 7 9 4 10 6 10 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 1 2 6 2 5 6 9 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 13 7 8 11 14 12 8 10 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 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 L11a546

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