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

K11n6 K11n8

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X18,12,19,11 X13,6,14,7 X20,16,21,15 X12,18,13,17 X22,20,1,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 6t-2 + 16t-1 - 21 + 16t - 6t2 + t3

Conway Polynomial:

1 + z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n131, K11n160, ...}

Determinant and Signature:

{67, 2}

Jones Polynomial:

q-3 - 3q-2 + 6q-1 - 9 + 11q - 11q2 + 11q3 - 8q4 + 5q5 - 2q6

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

{K11n36, K11n44, ...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 2q-4 - 2q-2 + q2 - 2q4 + 3q6 - q8 + 3q10 + q12 - q14 + 2q16 - 2q18 - q20

HOMFLY-PT Polynomial:

- a-6 + a-4 - a-4z4 + 2a-2 + 4a-2z2 + 3a-2z4 + a-2z6 - 2 - 4z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- 2a-7z + 3a-7z3 + a-6 - 3a-6z2 + 4a-6z4 + a-6z6 - 3a-5z + 5a-5z3 - 2a-5z5 + 3a-5z7 + a-4 - a-4z4 - a-4z6 + 3a-4z8 - 3a-3z + 11a-3z3 - 17a-3z5 + 7a-3z7 + a-3z9 - 2a-2 + 9a-2z2 - 9a-2z4 - 7a-2z6 + 6a-2z8 - 4a-1z + 17a-1z3 - 24a-1z5 + 7a-1z7 + a-1z9 - 2 + 9z2 - 7z4 - 4z6 + 3z8 - 2az + 8az3 - 9az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{1, 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=2 is the signature of 117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   2 j = 11                 3   j = 9               5 2   j = 7             6 3     j = 5           5 5       j = 3         6 6         j = 1       4 6           j = -1     2 5             j = -3   1 4               j = -5   2                 j = -7 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, NonAlternating, 7]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 7]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 14, 8, 15], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[13, 6, 14, 7], X[20, 16, 21, 15], > X[12, 18, 13, 17], X[22, 20, 1, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 7]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 7, -4, -2, 5, -3, 6, -9, -7, 4, 8, -11, 9, -6, 10, > -8, 11, -10]
In[5]:=	DTCode[Knot[11, NonAlternating, 7]]
Out[5]=	DTCode[4, 8, 10, -14, 2, 18, -6, 20, 12, 22, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 7]][t]
Out[6]=	-3 6 16 2 3 -21 + t - -- + -- + 16 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 7]][z]
Out[7]=	2 6 1 + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 7], Knot[11, NonAlternating, 131], > Knot[11, NonAlternating, 160]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 7]], KnotSignature[Knot[11, NonAlternating, 7]]}
Out[9]=	{67, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 7]][q]
Out[10]=	-3 3 6 2 3 4 5 6 -9 + q - -- + - + 11 q - 11 q + 11 q - 8 q + 5 q - 2 q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 7], Knot[11, NonAlternating, 36], > Knot[11, NonAlternating, 44]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 7]][q]
Out[12]=	-10 -6 2 2 2 4 6 8 10 12 14 16 q - q + -- - -- + q - 2 q + 3 q - q + 3 q + q - q + 2 q - 4 2 q q 18 20 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 7]][a, z]
Out[13]=	2 4 4 6 -6 -4 2 2 2 4 z 2 2 4 z 3 z z -2 - a + a + -- + a - 4 z + ---- + a z - 2 z - -- + ---- + -- 2 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 7]][a, z]
Out[14]=	2 2 -6 -4 2 2 2 z 3 z 3 z 4 z 2 3 z 9 z -2 + a + a - -- - a - --- - --- - --- - --- - 2 a z + 9 z - ---- + ---- + 2 7 5 3 a 6 2 a a a a a a 3 3 3 3 4 4 4 2 2 3 z 5 z 11 z 17 z 3 4 4 z z 9 z > 3 a z + ---- + ---- + ----- + ----- + 8 a z - 7 z + ---- - -- - ---- - 7 5 3 a 6 4 2 a a a a a a 5 5 5 6 6 6 2 4 2 z 17 z 24 z 5 6 z z 7 z 2 6 > 3 a z - ---- - ----- - ----- - 9 a z - 4 z + -- - -- - ---- + a z + 5 3 a 6 4 2 a a a a a 7 7 7 8 8 9 9 3 z 7 z 7 z 7 8 3 z 6 z z z > ---- + ---- + ---- + 3 a z + 3 z + ---- + ---- + -- + -- 5 3 a 4 2 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 7]], Vassiliev[3][Knot[11, NonAlternating, 7]]}
Out[15]=	{1, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 7]][q, t]
Out[16]=	3 1 2 1 4 2 5 4 q 3 6 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 6 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 5 q t + 5 q t + 6 q t + 3 q t + 5 q t + 2 q t + 3 q t + 13 5 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n7

K11n6 K11n8