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

K11a71 K11a73

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 2 22 18 20 8 6 16

Alexander Polynomial:

t-4 - 6t-3 + 18t-2 - 32t-1 + 39 - 32t + 18t2 - 6t3 + t4

Conway Polynomial:

1 + 2z2 + 2z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{153, 0}

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 15q-2 - 21q-1 + 25 - 24q + 22q2 - 16q3 + 10q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 3q-10 + 2q-8 + q-6 - 4q-4 + 5q-2 - 4 + 4q2 + q4 + 5q8 - 4q10 + q12 - q14 - 2q16 + q18

HOMFLY-PT Polynomial:

- a-4 + a-4z2 + a-4z4 + a-2 - 4a-2z2 - 6a-2z4 - 2a-2z6 + 2 + 8z2 + 10z4 + 5z6 + z8 - a2 - 3a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

- a-6z4 + a-6z6 + a-5z + 4a-5z3 - 10a-5z5 + 5a-5z7 - a-4 - 2a-4z2 + 12a-4z4 - 20a-4z6 + 9a-4z8 - a-3z + 14a-3z3 - 18a-3z5 - 4a-3z7 + 7a-3z9 - a-2 - 12a-2z2 + 44a-2z4 - 55a-2z6 + 19a-2z8 + 2a-2z10 - 5a-1z + 22a-1z3 - 15a-1z5 - 16a-1z7 + 14a-1z9 + 2 - 16z2 + 47z4 - 53z6 + 20z8 + 2z10 - 5az + 20az3 - 20az5 + az7 + 7az9 + a2 - 5a2z2 + 11a2z4 - 15a2z6 + 10a2z8 - 2a3z + 7a3z3 - 12a3z5 + 8a3z7 + a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

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

{2, 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 1172. 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                     4   j = 9                   6 1   j = 7                 10 4     j = 5               12 6       j = 3             12 10         j = 1           13 12           j = -1         9 13             j = -3       6 12               j = -5     3 9                 j = -7   1 6                   j = -9   3                     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, 72]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 72]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[14, 8, 15, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 14, 19, 13], X[20, 15, 21, 16], > X[8, 18, 9, 17], X[6, 19, 7, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 72]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 72]]
Out[5]=	DTCode[4, 10, 12, 14, 2, 22, 18, 20, 8, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 72]][t]
Out[6]=	-4 6 18 32 2 3 4 39 + t - -- + -- - -- - 32 t + 18 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 72]][z]
Out[7]=	2 4 6 8 1 + 2 z + 2 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 72]}
In[9]:=	{KnotDet[Knot[11, Alternating, 72]], KnotSignature[Knot[11, Alternating, 72]]}
Out[9]=	{153, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 72]][q]
Out[10]=	-5 4 9 15 21 2 3 4 5 6 25 - q + -- - -- + -- - -- - 24 q + 22 q - 16 q + 10 q - 5 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, 72]}
In[12]:=	A2Invariant[Knot[11, Alternating, 72]][q]
Out[12]=	-14 2 3 2 -6 4 5 2 4 8 10 12 -4 - q + --- - --- + -- + q - -- + -- + 4 q + q + 5 q - 4 q + q - 12 10 8 4 2 q q q q q 14 16 18 > q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 72]][a, z]
Out[13]=	2 2 4 4 -4 -2 2 2 z 4 z 2 2 4 z 6 z 2 4 2 - a + a - a + 8 z + -- - ---- - 3 a z + 10 z + -- - ---- - 3 a z + 4 2 4 2 a a a a 6 6 2 z 2 6 8 > 5 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 72]][a, z]
Out[14]=	2 2 -4 -2 2 z z 5 z 3 2 2 z 12 z 2 - a - a + a + -- - -- - --- - 5 a z - 2 a z - 16 z - ---- - ----- - 5 3 a 4 2 a a a a 3 3 3 2 2 4 2 4 z 14 z 22 z 3 3 3 5 3 > 5 a z + a z + ---- + ----- + ----- + 20 a z + 7 a z - a z + 5 3 a a a 4 4 4 5 5 5 4 z 12 z 44 z 2 4 4 4 10 z 18 z 15 z > 47 z - -- + ----- + ----- + 11 a z - 5 a z - ----- - ----- - ----- - 6 4 2 5 3 a a a a a a 6 6 6 5 3 5 5 5 6 z 20 z 55 z 2 6 > 20 a z - 12 a z + a z - 53 z + -- - ----- - ----- - 15 a z + 6 4 2 a a a 7 7 7 8 8 4 6 5 z 4 z 16 z 7 3 7 8 9 z 19 z > 4 a z + ---- - ---- - ----- + a z + 8 a z + 20 z + ---- + ----- + 5 3 a 4 2 a a a a 9 9 10 2 8 7 z 14 z 9 10 2 z > 10 a z + ---- + ----- + 7 a z + 2 z + ----- 3 a 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 72]], Vassiliev[3][Knot[11, Alternating, 72]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 72]][q, t]
Out[16]=	13 1 3 1 6 3 9 6 12 -- + 13 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 9 3 3 2 5 2 5 3 7 3 > --- + 12 q t + 12 q t + 10 q t + 12 q t + 6 q t + 10 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a72

K11a71 K11a73