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

K11n71 K11n73

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X20,10,21,9 X16,12,17,11 X13,6,14,7 X18,16,19,15 X12,18,13,17 X22,20,1,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 -14 2 20 16 -6 18 12 22 10

Alexander Polynomial:

- 2t-3 + 9t-2 - 18t-1 + 23 - 18t + 9t2 - 2t3

Conway Polynomial:

1 - 3z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1087, 1098, K11a58, K11a165, ...}

Determinant and Signature:

{81, 4}

Jones Polynomial:

3q2 - 5q3 + 10q4 - 13q5 + 13q6 - 14q7 + 11q8 - 7q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

3q6 + 5q10 + 2q12 - 2q14 - 7q18 - q20 - 2q22 + q24 + 4q26 - q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 6a-8 + 8a-8z2 + 3a-8z4 - 11a-6 - 16a-6z2 - 9a-6z4 - 2a-6z6 + 7a-4 + 9a-4z2 + 3a-4z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + 3a-12z2 - 7a-12z4 + 4a-12z6 - 3a-11z + 5a-11z3 - 10a-11z5 + 6a-11z7 + a-10 + 11a-10z2 - 19a-10z4 + 3a-10z6 + 4a-10z8 - 15a-9z + 37a-9z3 - 38a-9z5 + 14a-9z7 + a-9z9 + 6a-8 - 3a-8z2 + 2a-8z4 - 10a-8z6 + 8a-8z8 - 21a-7z + 42a-7z3 - 33a-7z5 + 11a-7z7 + a-7z9 + 11a-6 - 24a-6z2 + 20a-6z4 - 9a-6z6 + 4a-6z8 - 9a-5z + 11a-5z3 - 6a-5z5 + 3a-5z7 + 7a-4 - 13a-4z2 + 6a-4z4

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

{0, -3}

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=4 is the signature of 1172. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   1 j = 21                 3   j = 19               4 1   j = 17             7 3     j = 15           7 4       j = 13         6 7         j = 11       7 7           j = 9     3 6             j = 7   2 7               j = 5 1 3                 j = 3 3

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, NonAlternating, 72]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 72]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[16, 12, 17, 11], X[13, 6, 14, 7], X[18, 16, 19, 15], > X[12, 18, 13, 17], X[22, 20, 1, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 72]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -11, 6, -9, -7, 3, 8, -6, 9, -8, 10, > -5, 11, -10]
In[5]:=	DTCode[Knot[11, NonAlternating, 72]]
Out[5]=	DTCode[4, 8, -14, 2, 20, 16, -6, 18, 12, 22, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 72]][t]
Out[6]=	2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 72]][z]
Out[7]=	4 6 1 - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 72]], KnotSignature[Knot[11, NonAlternating, 72]]}
Out[9]=	{81, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 72]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 3 q - 5 q + 10 q - 13 q + 13 q - 14 q + 11 q - 7 q + 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 72]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 72]][q]
Out[12]=	6 10 12 14 18 20 22 24 26 28 3 q + 5 q + 2 q - 2 q - 7 q - q - 2 q + q + 4 q - q + 30 34 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 72]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 6 11 7 z 8 z 16 z 9 z 3 z 9 z 3 z 2 z -a + -- - -- + -- - --- + ---- - ----- + ---- + ---- - ---- + ---- - ---- 8 6 4 10 8 6 4 8 6 4 6 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 72]][a, z]
Out[14]=	2 2 2 2 -10 6 11 7 3 z 15 z 21 z 9 z 3 z 11 z 3 z 24 z a + -- + -- + -- - --- - ---- - ---- - --- + ---- + ----- - ---- - ----- - 8 6 4 11 9 7 5 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 13 z z 5 z 37 z 42 z 11 z 7 z 19 z 2 z 20 z > ----- - --- + ---- + ----- + ----- + ----- - ---- - ----- + ---- + ----- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 6 z z 10 z 38 z 33 z 6 z 4 z 3 z 10 z 9 z > ---- + --- - ----- - ----- - ----- - ---- + ---- + ---- - ----- - ---- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 6 z 14 z 11 z 3 z 4 z 8 z 4 z z z > ---- + ----- + ----- + ---- + ---- + ---- + ---- + -- + -- 11 9 7 5 10 8 6 9 7 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 72]], Vassiliev[3][Knot[11, NonAlternating, 72]]}
Out[15]=	{0, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 72]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 3 q + q + 3 q t + 2 q t + 7 q t + 3 q t + 6 q t + 7 q t + 11 4 13 4 13 5 15 5 15 6 17 6 > 7 q t + 6 q t + 7 q t + 7 q t + 4 q t + 7 q t + 17 7 19 7 19 8 21 8 23 9 > 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n72

K11n71 K11n73