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DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X16,11,5,12 X4,15,1,16 X20,14,21,13 X18,7,19,8 X8,17,9,18 X22,20,17,19 X14,22,15,21

Gauss Code:

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

Jones Polynomial:

q-9 - 3q-8 + 7q-7 - 13q-6 + 19q-5 - 22q-4 + 25q-3 - 20q-2 + 17q-1 - 11 + 5q - q2

A2 (sl(3)) Invariant:

q-28 - q-24 + 3q-22 - 3q-20 - q-18 + 6q-16 + 8q-12 + 4q-10 + 4q-8 + 6q-6 - 3q-4 + 5q-2 - 2 - 2q2 + 3q4 - q6

HOMFLY-PT Polynomial:

- z4 + a2z-2 + 2a2 + a2z2 + a2z4 + a2z6 - 2a4z-2 - a4 + a4z2 + a4z4 + a4z6 + a6z-2 - 2a6 - 3a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

a-1z5 - 4z4 + 5z6 + 2az3 - 15az5 + 11az7 + a2z-2 - 2a2 + a2z2 + 7a2z4 - 20a2z6 + 13a2z8 - 2a3z-1 + 2a3z + 4a3z3 - 12a3z5 - 2a3z7 + 8a3z9 + 2a4z-2 - 2a4 - a4z2 + 23a4z4 - 39a4z6 + 17a4z8 + 2a4z10 - 2a5z-1 - 2a5z + 8a5z3 + a5z5 - 17a5z7 + 12a5z9 + a6z-2 - a6 + a6z2 + 11a6z4 - 20a6z6 + 8a6z8 + 2a6z10 - 6a7z + 13a7z3 - 11a7z5 - a7z7 + 4a7z9 - a8 + 6a8z2 - 4a8z4 - 5a8z6 + 4a8z8 - 2a9z + 7a9z3 - 8a9z5 + 3a9z7 - a10 + 3a10z2 - 3a10z4 + a10z6

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 = 5                       1 j = 3                     4   j = 1                   7 1   j = -1                 10 4     j = -3               12 9       j = -5             13 8         j = -7           9 12           j = -9         10 13             j = -11       5 11               j = -13     2 8                 j = -15   1 5                   j = -17   2                     j = -19 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 482]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 482]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[10, 5, 11, 6], > X[16, 11, 5, 12], X[4, 15, 1, 16], X[20, 14, 21, 13], X[18, 7, 19, 8], > X[8, 17, 9, 18], X[22, 20, 17, 19], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6}, {9, -8, 10, -7, 11, -10}, > {4, -1, 8, -9, 2, -4, 5, -3, 7, -11, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-9 3 7 13 19 22 25 20 17 2 -11 + q - -- + -- - -- + -- - -- + -- - -- + -- + 5 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -24 3 3 -18 6 8 4 4 6 3 5 -2 + q - q + --- - --- - q + --- + --- + --- + -- + -- - -- + -- - 22 20 16 12 10 8 6 4 2 q q q q q q q q q 2 4 6 > 2 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 482]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 4 2 6 2 8 2 4 2 a - a - 2 a + a + -- - ---- + -- + a z + a z - 3 a z + a z - z + 2 2 2 z z z 2 4 4 4 6 4 2 6 4 6 > a z + a z - 2 a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 482]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 5 -2 a - 2 a - a - a - a + -- + ---- + -- - ---- - ---- + 2 a z - 2 a z - 2 2 2 z z z z z 7 9 2 2 4 2 6 2 8 2 10 2 3 > 6 a z - 2 a z + a z - a z + a z + 6 a z + 3 a z + 2 a z + 3 3 5 3 7 3 9 3 4 2 4 4 4 > 4 a z + 8 a z + 13 a z + 7 a z - 4 z + 7 a z + 23 a z + 5 6 4 8 4 10 4 z 5 3 5 5 5 > 11 a z - 4 a z - 3 a z + -- - 15 a z - 12 a z + a z - a 7 5 9 5 6 2 6 4 6 6 6 8 6 > 11 a z - 8 a z + 5 z - 20 a z - 39 a z - 20 a z - 5 a z + 10 6 7 3 7 5 7 7 7 9 7 2 8 > a z + 11 a z - 2 a z - 17 a z - a z + 3 a z + 13 a z + 4 8 6 8 8 8 3 9 5 9 7 9 4 10 > 17 a z + 8 a z + 4 a z + 8 a z + 12 a z + 4 a z + 2 a z + 6 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 10 1 2 1 5 2 8 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 11 10 13 9 12 13 8 12 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a482

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