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

K11n70 K11n72

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X9,20,10,21 X16,12,17,11 X6,14,7,13 X18,16,19,15 X12,18,13,17 X19,22,20,1 X21,10,22,11

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 - 7t-2 + 14t-1 - 17 + 14t - 7t2 + 2t3

Conway Polynomial:

1 + 4z2 + 5z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1065, 1077, K11n75, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

- 2 + 5q - 7q2 + 11q3 - 10q4 + 10q5 - 9q6 + 5q7 - 3q8 + q9

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

{K11n75, ...}

A2 (sl(3)) Invariant:

- 2 + q2 - 2q4 + q6 + 4q8 + 2q10 + 6q12 - 3q18 - 5q20 - q24 + q26 + q28

HOMFLY-PT Polynomial:

2a-8 + a-8z2 - 9a-6 - 10a-6z2 - 3a-6z4 + 11a-4 + 18a-4z2 + 10a-4z4 + 2a-4z6 - 3a-2 - 5a-2z2 - 2a-2z4

Kauffman Polynomial:

2a-10z2 - 3a-10z4 + a-10z6 - 4a-9z + 10a-9z3 - 10a-9z5 + 3a-9z7 + 2a-8 + a-8z2 - a-8z4 - 6a-8z6 + 3a-8z8 - 17a-7z + 36a-7z3 - 31a-7z5 + 7a-7z7 + a-7z9 + 9a-6 - 15a-6z2 + 17a-6z4 - 18a-6z6 + 7a-6z8 - 21a-5z + 42a-5z3 - 31a-5z5 + 8a-5z7 + a-5z9 + 11a-4 - 20a-4z2 + 19a-4z4 - 10a-4z6 + 4a-4z8 - 11a-3z + 19a-3z3 - 10a-3z5 + 4a-3z7 + 3a-2 - 6a-2z2 + 4a-2z4 + a-2z6 - 3a-1z + 3a-1z3

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

{4, 5}

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=2 is the signature of 1171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

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, 71]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 71]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[9, 20, 10, 21], X[16, 12, 17, 11], X[6, 14, 7, 13], X[18, 16, 19, 15], > X[12, 18, 13, 17], X[19, 22, 20, 1], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 71]]
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, 71]]
Out[5]=	DTCode[4, 8, 14, 2, -20, 16, 6, 18, 12, -22, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 71]][t]
Out[6]=	2 7 14 2 3 -17 + -- - -- + -- + 14 t - 7 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 71]][z]
Out[7]=	2 4 6 1 + 4 z + 5 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 65], Knot[10, 77], Knot[11, NonAlternating, 71], > Knot[11, NonAlternating, 75]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 71]], KnotSignature[Knot[11, NonAlternating, 71]]}
Out[9]=	{63, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 71]][q]
Out[10]=	2 3 4 5 6 7 8 9 -2 + 5 q - 7 q + 11 q - 10 q + 10 q - 9 q + 5 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 71], Knot[11, NonAlternating, 75]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 71]][q]
Out[12]=	2 4 6 8 10 12 18 20 24 26 28 -2 + q - 2 q + q + 4 q + 2 q + 6 q - 3 q - 5 q - q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 71]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 2 9 11 3 z 10 z 18 z 5 z 3 z 10 z 2 z 2 z -- - -- + -- - -- + -- - ----- + ----- - ---- - ---- + ----- - ---- + ---- 8 6 4 2 8 6 4 2 6 4 2 4 a a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 71]][a, z]
Out[14]=	2 2 2 2 9 11 3 4 z 17 z 21 z 11 z 3 z 2 z z 15 z -- + -- + -- + -- - --- - ---- - ---- - ---- - --- + ---- + -- - ----- - 8 6 4 2 9 7 5 3 a 10 8 6 a a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 20 z 6 z 10 z 36 z 42 z 19 z 3 z 3 z z 17 z > ----- - ---- + ----- + ----- + ----- + ----- + ---- - ---- - -- + ----- + 4 2 9 7 5 3 a 10 8 6 a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 19 z 4 z 10 z 31 z 31 z 10 z z 6 z 18 z 10 z > ----- + ---- - ----- - ----- - ----- - ----- + --- - ---- - ----- - ----- + 4 2 9 7 5 3 10 8 6 4 a a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 z 3 z 7 z 8 z 4 z 3 z 7 z 4 z z z > -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 2 9 7 5 3 8 6 4 7 5 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 71]], Vassiliev[3][Knot[11, NonAlternating, 71]]}
Out[15]=	{4, 5}
In[16]:=	Kh[Knot[11, NonAlternating, 71]][q, t]
Out[16]=	3 2 3 5 5 2 7 2 7 3 9 3 3 q + 3 q + --- + 5 q t + 2 q t + 6 q t + 5 q t + 4 q t + 6 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 6 q t + 4 q t + 3 q t + 6 q t + 2 q t + 3 q t + q t + 17 7 19 8 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n71

K11n70 K11n72