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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 15q-9/2 + 20q-7/2 - 27q-5/2 + 25q-3/2 - 25q-1/2 + 18q1/2 - 13q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + 2q-16 + 9q-14 + 3q-12 + 13q-10 + 10q-8 + 9q-6 + 14q-4 + 4q-2 + 13 + 2q2 + 2q4 + 5q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z-3 - 3a-1z-1 - 2a-1z - a-1z3 - a-1z5 + 3az-3 + 9az-1 + 9az + 6az3 + 3az5 + az7 - 3a3z-3 - 10a3z-1 - 12a3z - 8a3z3 - 3a3z5 + a5z-3 + 5a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- a-3z5 + 2a-2z4 - 5a-2z6 + a-1z-3 - 5a-1z-1 + 7a-1z - 9a-1z3 + 18a-1z5 - 13a-1z7 - 3z-2 + 9 - 5z2 - 5z4 + 23z6 - 16z8 + 3az-3 - 14az-1 + 30az - 43az3 + 43az5 - 7az7 - 9az9 - 6a2z-2 + 21a2 - 24a2z2 - 11a2z4 + 48a2z6 - 24a2z8 - 2a2z10 + 3a3z-3 - 18a3z-1 + 48a3z - 68a3z3 + 44a3z5 + 5a3z7 - 13a3z9 - 3a4z-2 + 18a4 - 30a4z2 + 2a4z4 + 24a4z6 - 12a4z8 - 2a4z10 + a5z-3 - 11a5z-1 + 31a5z - 43a5z3 + 28a5z5 - 4a5z7 - 4a5z9 + 6a6 - 14a6z2 + 9a6z4 + 3a6z6 - 4a6z8 - 2a7z-1 + 6a7z - 9a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     4   j = 4                   9 1   j = 2                 9 4     j = 0               16 9       j = -2             15 15         j = -4           12 10           j = -6         8 15             j = -8       7 12               j = -10     2 10                 j = -12   1 5                   j = -14   2                     j = -16 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, 541]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 541]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[20, 12, 21, 11], X[22, 13, 19, 14], > X[18, 22, 9, 21], X[12, 17, 13, 18], X[8, 16, 5, 15], X[14, 8, 15, 7], > X[16, 19, 17, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 8, -7}, {9, -3, 5, -4}, > {11, -2, 3, -6, 4, -8, 7, -9, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 15 20 27 25 25 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 18 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 13 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 2 9 3 13 10 9 14 4 13 - q - q + q - --- + --- + --- + --- + --- + -- + -- + -- + -- + 18 16 14 12 10 8 6 4 2 q q q q q q q q q 2 4 6 8 10 > 2 q + 2 q + 5 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 541]][a, z]
Out[8]=	3 5 3 5 7 1 3 a 3 a a 3 9 a 10 a 5 a a 2 z -(----) + --- - ---- + -- - --- + --- - ----- + ---- - -- - --- + 9 a z - 3 3 3 3 a z z z z z a a z z z z 3 5 3 5 7 z 3 3 3 5 3 z 5 > 12 a z + 6 a z - a z - -- + 6 a z - 8 a z + 3 a z - -- + 3 a z - a a 3 5 7 > 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 541]][a, z]
Out[9]=	3 5 2 4 2 4 6 8 1 3 a 3 a a 3 6 a 3 a 9 + 21 a + 18 a + 6 a + a + ---- + --- + ---- + -- - -- - ---- - ---- - 3 3 3 3 2 2 2 a z z z z z z z 3 5 7 5 14 a 18 a 11 a 2 a 7 z 3 5 > --- - ---- - ----- - ----- - ---- + --- + 30 a z + 48 a z + 31 a z + a z z z z z a 3 7 2 2 2 4 2 6 2 8 2 9 z 3 > 6 a z - 5 z - 24 a z - 30 a z - 14 a z - 3 a z - ---- - 43 a z - a 4 3 3 5 3 7 3 4 2 z 2 4 4 4 > 68 a z - 43 a z - 9 a z - 5 z + ---- - 11 a z + 2 a z + 2 a 5 5 6 4 8 4 z 18 z 5 3 5 5 5 7 5 > 9 a z + 3 a z - -- + ----- + 43 a z + 44 a z + 28 a z + 8 a z + 3 a a 6 7 6 5 z 2 6 4 6 6 6 8 6 13 z 7 > 23 z - ---- + 48 a z + 24 a z + 3 a z - a z - ----- - 7 a z + 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 5 a z - 4 a z - 3 a z - 16 z - 24 a z - 12 a z - 4 a z - 9 3 9 5 9 2 10 4 10 > 9 a z - 13 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	15 1 2 1 5 2 10 7 12 16 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 15 12 10 15 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 9 t + 9 q t + 4 q t + 9 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a541

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