© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a87 K11a89

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X22,16,1,15 X8,17,9,18 X6,19,7,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 18 2 20 22 8 6 14

Alexander Polynomial:

t-4 - 5t-3 + 12t-2 - 20t-1 + 25 - 20t + 12t2 - 5t3 + t4

Conway Polynomial:

1 - z2 + 2z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{101, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 10q-3 + 14q-2 - 16q-1 + 16 - 14q + 11q2 - 6q3 + 3q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a84, ...}

A2 (sl(3)) Invariant:

q-18 + q-12 - 3q-10 + 2q-8 - q-6 - q-4 + 2q-2 - 3 + 4q2 - q4 + 2q6 + 2q8 - 2q10 + q12 - q14

HOMFLY-PT Polynomial:

- a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + 5 + 14z2 + 14z4 + 6z6 + z8 - 5a2 - 13a2z2 - 9a2z4 - 2a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 3a-4z6 - a-3z + 6a-3z3 - 10a-3z5 + 5a-3z7 + a-2 - 7a-2z2 + 16a-2z4 - 14a-2z6 + 6a-2z8 - a-1z + 3a-1z3 + 2a-1z5 - 5a-1z7 + 4a-1z9 + 5 - 26z2 + 45z4 - 31z6 + 9z8 + z10 + az - 10az3 + 18az5 - 16az7 + 7az9 + 5a2 - 23a2z2 + 33a2z4 - 25a2z6 + 7a2z8 + a2z10 + 2a3z3 - 4a3z5 - 3a3z7 + 3a3z9 + 2a4 - 4a4z2 + 7a4z4 - 10a4z6 + 4a4z8 - a5z + 7a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 1188. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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 = 11                       1 j = 9                     2   j = 7                   4 1   j = 5                 7 2     j = 3               7 4       j = 1             9 7         j = -1           8 8           j = -3         6 8             j = -5       4 8               j = -7     2 6                 j = -9   1 4                   j = -11   2                     j = -13 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 88]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 88]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[8, 17, 9, 18], X[6, 19, 7, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 88]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 88]]
Out[5]=	DTCode[4, 10, 12, 16, 18, 2, 20, 22, 8, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 88]][t]
Out[6]=	-4 5 12 20 2 3 4 25 + t - -- + -- - -- - 20 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 88]][z]
Out[7]=	2 4 6 8 1 - z + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 88]}
In[9]:=	{KnotDet[Knot[11, Alternating, 88]], KnotSignature[Knot[11, Alternating, 88]]}
Out[9]=	{101, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 88]][q]
Out[10]=	-6 3 6 10 14 16 2 3 4 5 16 + q - -- + -- - -- + -- - -- - 14 q + 11 q - 6 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 84], Knot[11, Alternating, 88]}
In[12]:=	A2Invariant[Knot[11, Alternating, 88]][q]
Out[12]=	-18 -12 3 2 -6 -4 2 2 4 6 8 -3 + q + q - --- + -- - q - q + -- + 4 q - q + 2 q + 2 q - 10 8 2 q q q 10 12 14 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 88]][a, z]
Out[13]=	2 4 -2 2 4 2 5 z 2 2 4 2 4 4 z 5 - a - 5 a + 2 a + 14 z - ---- - 13 a z + 3 a z + 14 z - ---- - 2 2 a a 6 2 4 4 4 6 z 2 6 8 > 9 a z + a z + 6 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 88]][a, z]
Out[14]=	2 2 -2 2 4 z z 5 2 2 z 7 z 2 2 5 + a + 5 a + 2 a - -- - - + a z - a z - 26 z + ---- - ---- - 23 a z - 3 a 4 2 a a a 3 3 3 4 2 6 2 2 z 6 z 3 z 3 3 3 5 3 > 4 a z + 2 a z - ---- + ---- + ---- - 10 a z + 2 a z + 7 a z + 5 3 a a a 4 4 5 5 5 4 6 z 16 z 2 4 4 4 6 4 z 10 z 2 z > 45 z - ---- + ----- + 33 a z + 7 a z - 3 a z + -- - ----- + ---- + 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 14 z 2 6 4 6 > 18 a z - 4 a z - 9 a z - 31 z + ---- - ----- - 25 a z - 10 a z + 4 2 a a 7 7 8 6 6 5 z 5 z 7 3 7 5 7 8 6 z 2 8 > a z + ---- - ---- - 16 a z - 3 a z + 3 a z + 9 z + ---- + 7 a z + 3 a 2 a a 9 4 8 4 z 9 3 9 10 2 10 > 4 a z + ---- + 7 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 88]], Vassiliev[3][Knot[11, Alternating, 88]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 88]][q, t]
Out[16]=	8 1 2 1 4 2 6 4 8 - + 9 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 8 8 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 7 q t + 4 q t + 7 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a88

K11a87 K11a89