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

K11a167 K11a169

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 16 14 2 20 8 22 12 6

Alexander Polynomial:

- 2t-3 + 12t-2 - 29t-1 + 39 - 29t + 12t2 - 2t3

Conway Polynomial:

1 + z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a67, K11a104, ...}

Determinant and Signature:

{125, 0}

Jones Polynomial:

- q-5 + 3q-4 - 7q-3 + 13q-2 - 17q-1 + 20 - 20q + 18q2 - 13q3 + 8q4 - 4q5 + q6

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

{K11a104, ...}

A2 (sl(3)) Invariant:

- q-16 + q-12 - 3q-10 + 3q-8 + 2q-6 - 2q-4 + 4q-2 - 3 + 2q2 - q6 + 4q8 - 4q10 + q12 + q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 - 2a-2z2 - 2a-2z4 - a-2z6 - z4 - z6 + 2a2 + 3a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 5a-5z3 - 10a-5z5 + 4a-5z7 - 4a-4z2 + 15a-4z4 - 19a-4z6 + 7a-4z8 + 2a-3z3 + 2a-3z5 - 11a-3z7 + 6a-3z9 - 10a-2z2 + 32a-2z4 - 33a-2z6 + 9a-2z8 + 2a-2z10 - 6a-1z3 + 18a-1z5 - 23a-1z7 + 11a-1z9 - 6z2 + 20z4 - 22z6 + 8z8 + 2z10 - az5 - 3az7 + 5az9 - 2a2 + 3a2z2 - 6a2z6 + 6a2z8 + a3z + a3z3 - 6a3z5 + 5a3z7 - a4 + 3a4z2 - 5a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{1, -1}

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 11168. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   5 1   j = 7                 8 3     j = 5               10 5       j = 3             10 8         j = 1           10 10           j = -1         8 11             j = -3       5 9               j = -5     2 8                 j = -7   1 5                   j = -9   2                     j = -11 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, 168]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 168]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[16, 8, 17, 7], > X[14, 10, 15, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[8, 16, 9, 15], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 168]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -8, 5, -2, 6, -10, 7, -5, 8, -4, 9, -3, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 168]]
Out[5]=	DTCode[4, 10, 18, 16, 14, 2, 20, 8, 22, 12, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 168]][t]
Out[6]=	2 12 29 2 3 39 - -- + -- - -- - 29 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 168]][z]
Out[7]=	2 6 1 + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 67], Knot[11, Alternating, 104], > Knot[11, Alternating, 168]}
In[9]:=	{KnotDet[Knot[11, Alternating, 168]], KnotSignature[Knot[11, Alternating, 168]]}
Out[9]=	{125, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 168]][q]
Out[10]=	-5 3 7 13 17 2 3 4 5 6 20 - q + -- - -- + -- - -- - 20 q + 18 q - 13 q + 8 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 104], Knot[11, Alternating, 168]}
In[12]:=	A2Invariant[Knot[11, Alternating, 168]][q]
Out[12]=	-16 -12 3 3 2 2 4 2 6 8 10 12 -3 - q + q - --- + -- + -- - -- + -- + 2 q - q + 4 q - 4 q + q + 10 8 6 4 2 q q q q q 14 16 18 > q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 168]][a, z]
Out[13]=	2 2 4 4 6 2 4 z 2 z 2 2 4 2 4 z 2 z 2 4 6 z 2 a - a + -- - ---- + 3 a z - a z - z + -- - ---- + 2 a z - z - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 168]][a, z]
Out[14]=	2 2 3 2 4 3 5 2 4 z 10 z 2 2 4 2 5 z -2 a - a + a z + a z - 6 z - ---- - ----- + 3 a z + 3 a z + ---- + 4 2 5 a a a 3 3 4 4 4 2 z 6 z 3 3 5 3 4 2 z 15 z 32 z 4 4 > ---- - ---- + a z - 2 a z + 20 z - ---- + ----- + ----- - 5 a z - 3 a 6 4 2 a a a a 5 5 5 6 6 10 z 2 z 18 z 5 3 5 5 5 6 z 19 z > ----- + ---- + ----- - a z - 6 a z + a z - 22 z + -- - ----- - 5 3 a 6 4 a a a a 6 7 7 7 33 z 2 6 4 6 4 z 11 z 23 z 7 3 7 > ----- - 6 a z + 3 a z + ---- - ----- - ----- - 3 a z + 5 a z + 2 5 3 a a a a 8 8 9 9 10 8 7 z 9 z 2 8 6 z 11 z 9 10 2 z > 8 z + ---- + ---- + 6 a z + ---- + ----- + 5 a z + 2 z + ----- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 168]], Vassiliev[3][Knot[11, Alternating, 168]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, Alternating, 168]][q, t]
Out[16]=	11 1 2 1 5 2 8 5 9 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 7 4 > --- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a168

K11a167 K11a169