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L11a547 L11n1

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - q-8 + 6q-7 - 6q-6 + 16q-5 - 15q-4 + 21q-3 - 15q-2 + 15q-1 - 10 + 5q - q2

A2 (sl(3)) Invariant:

q-30 + 4q-28 + 7q-26 + 13q-24 + 23q-22 + 26q-20 + 31q-18 + 37q-16 + 28q-14 + 27q-12 + 19q-10 + 12q-8 + 11q-6 + 5q-2 - 1 - 2q2 + 3q4 - q6

HOMFLY-PT Polynomial:

- z4 + a2z-4 + 4a2z-2 + 6a2 + 4a2z2 + 2a2z4 + a2z6 - 4a4z-4 - 13a4z-2 - 16a4 - 10a4z2 - 4a4z4 + 6a6z-4 + 15a6z-2 + 14a6 + 6a6z2 - 4a8z-4 - 7a8z-2 - 4a8 + a10z-4 + a10z-2

Kauffman Polynomial:

a-1z5 - 5z4 + 5z6 - 15az5 + 10az7 - a2z-4 + 5a2z-2 - 10a2 + 10a2z2 - 15a2z6 + 10a2z8 + 4a3z-3 - 15a3z-1 + 20a3z - 10a3z3 - 5a3z5 + 5a3z9 - 4a4z-4 + 14a4z-2 - 25a4 + 30a4z2 - 11a4z4 - 14a4z6 + 10a4z8 + a4z10 + 12a5z-3 - 41a5z-1 + 55a5z - 30a5z3 + 11a5z5 - 8a5z7 + 6a5z9 - 6a6z-4 + 18a6z-2 - 31a6 + 40a6z2 - 31a6z4 + 9a6z6 + a6z8 + a6z10 + 12a7z-3 - 41a7z-1 + 55a7z - 30a7z3 + 3a7z7 + a7z9 - 4a8z-4 + 14a8z-2 - 25a8 + 30a8z2 - 20a8z4 + 4a8z6 + a8z8 + 4a9z-3 - 15a9z-1 + 20a9z - 10a9z3 + a9z7 - a10z-4 + 5a10z-2 - 10a10 + 10a10z2 - 5a10z4 + 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                   6 1   j = -1                 9 4     j = -3               11 11       j = -5             10 4         j = -7           5 11           j = -9         11 10             j = -11       5 15               j = -13     1 1                 j = -15     5                   j = -17 1 1                     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]=	5
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 548]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 548]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[22, 20, 17, 19], > X[16, 22, 13, 21], X[20, 16, 21, 15], X[12, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 10, -9}, > {11, -3, 8, -10, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-9 -8 6 6 16 15 21 15 15 2 -10 + q - q + -- - -- + -- - -- + -- - -- + -- + 5 q - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 4 7 13 23 26 31 37 28 27 19 12 -1 + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- + 28 26 24 22 20 18 16 14 12 10 8 q q q q q q q q q q q 11 5 2 4 6 > -- + -- - 2 q + 3 q - q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 548]][a, z]
Out[8]=	2 4 6 8 10 2 4 2 4 6 8 a 4 a 6 a 4 a a 4 a 13 a 6 a - 16 a + 14 a - 4 a + -- - ---- + ---- - ---- + --- + ---- - ----- + 4 4 4 4 4 2 2 z z z z z z z 6 8 10 15 a 7 a a 2 2 4 2 6 2 4 2 4 > ----- - ---- + --- + 4 a z - 10 a z + 6 a z - z + 2 a z - 2 2 2 z z z 4 4 2 6 > 4 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 548]][a, z]
Out[9]=	2 4 6 8 10 2 4 6 8 10 a 4 a 6 a 4 a a -10 a - 25 a - 31 a - 25 a - 10 a - -- - ---- - ---- - ---- - --- + 4 4 4 4 4 z z z z z 3 5 7 9 2 4 6 8 10 4 a 12 a 12 a 4 a 5 a 14 a 18 a 14 a 5 a > ---- + ----- + ----- + ---- + ---- + ----- + ----- + ----- + ----- - 3 3 3 3 2 2 2 2 2 z z z z z z z z z 3 5 7 9 15 a 41 a 41 a 15 a 3 5 7 9 > ----- - ----- - ----- - ----- + 20 a z + 55 a z + 55 a z + 20 a z + z z z z 2 2 4 2 6 2 8 2 10 2 3 3 > 10 a z + 30 a z + 40 a z + 30 a z + 10 a z - 10 a z - 5 3 7 3 9 3 4 4 4 6 4 8 4 > 30 a z - 30 a z - 10 a z - 5 z - 11 a z - 31 a z - 20 a z - 5 10 4 z 5 3 5 5 5 6 2 6 4 6 > 5 a z + -- - 15 a z - 5 a z + 11 a z + 5 z - 15 a z - 14 a z + a 6 6 8 6 10 6 7 5 7 7 7 9 7 > 9 a z + 4 a z + a z + 10 a z - 8 a z + 3 a z + a z + 2 8 4 8 6 8 8 8 3 9 5 9 7 9 4 10 > 10 a z + 10 a z + a z + a z + 5 a z + 6 a z + a z + a z + 6 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 9 1 1 1 5 1 1 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 8 17 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 15 11 10 5 11 10 4 11 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 > 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a548

L11a547 L11n1