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

K11n92 K11n94

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X7,13,8,12 X2,10,3,9 X11,21,12,20 X18,14,19,13 X22,16,1,15 X6,18,7,17 X19,9,20,8 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

3t-3 - 7t-2 + 9t-1 - 9 + 9t - 7t2 + 3t3

Conway Polynomial:

1 + 8z2 + 11z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a242, ...}

Determinant and Signature:

{47, 6}

Jones Polynomial:

q3 - 2q4 + 5q5 - 6q6 + 8q7 - 8q8 + 7q9 - 6q10 + 3q11 - q12

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + 2q14 + 2q18 + 2q20 + 3q24 - 2q26 - 2q30 - 2q32 - q36 + q38

HOMFLY-PT Polynomial:

a-12 - 5a-10 - 7a-10z2 - 2a-10z4 + 4a-8 + 11a-8z2 + 9a-8z4 + 2a-8z6 + a-6 + 4a-6z2 + 4a-6z4 + a-6z6

Kauffman Polynomial:

- a-15z + a-15z3 - 2a-14z2 + 3a-14z4 - a-13z + 2a-13z3 + a-13z7 + a-12 - 3a-12z2 + 8a-12z4 - 5a-12z6 + 2a-12z8 - 7a-11z + 15a-11z3 - 9a-11z5 + a-11z7 + a-11z9 + 5a-10 - 17a-10z2 + 23a-10z4 - 17a-10z6 + 5a-10z8 - 7a-9z + 17a-9z3 - 15a-9z5 + 2a-9z7 + a-9z9 + 4a-8 - 12a-8z2 + 14a-8z4 - 11a-8z6 + 3a-8z8 + 3a-7z3 - 6a-7z5 + 2a-7z7 - a-6 + 4a-6z2 - 4a-6z4 + a-6z6

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

{8, 21}

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=6 is the signature of 1193. 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 = 25                   1 j = 23                 2   j = 21               4 1   j = 19             3 2     j = 17           5 4       j = 15         3 3         j = 13       3 5           j = 11     2 3             j = 9     3               j = 7 1 2                 j = 5 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, 93]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 93]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[7, 13, 8, 12], > X[2, 10, 3, 9], X[11, 21, 12, 20], X[18, 14, 19, 13], X[22, 16, 1, 15], > X[6, 18, 7, 17], X[19, 9, 20, 8], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 93]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, -4, 10, 5, -2, -6, 4, 7, -3, 8, -11, 9, -7, -10, > 6, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 93]]
Out[5]=	DTCode[4, 10, 14, -12, 2, -20, 18, 22, 6, -8, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 93]][t]
Out[6]=	3 7 9 2 3 -9 + -- - -- + - + 9 t - 7 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 93]][z]
Out[7]=	2 4 6 1 + 8 z + 11 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 242], Knot[11, NonAlternating, 93]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 93]], KnotSignature[Knot[11, NonAlternating, 93]]}
Out[9]=	{47, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 93]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 5 q - 6 q + 8 q - 8 q + 7 q - 6 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 93]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 93]][q]
Out[12]=	10 12 14 18 20 24 26 30 32 36 38 q - q + 2 q + 2 q + 2 q + 3 q - 2 q - 2 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 93]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -12 5 4 -6 7 z 11 z 4 z 2 z 9 z 4 z 2 z z a - --- + -- + a - ---- + ----- + ---- - ---- + ---- + ---- + ---- + -- 10 8 10 8 6 10 8 6 8 6 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 93]][a, z]
Out[14]=	2 2 2 2 -12 5 4 -6 z z 7 z 7 z 2 z 3 z 17 z 12 z a + --- + -- - a - --- - --- - --- - --- - ---- - ---- - ----- - ----- + 10 8 15 13 11 9 14 12 10 8 a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 4 z z 2 z 15 z 17 z 3 z 3 z 8 z 23 z 14 z > ---- + --- + ---- + ----- + ----- + ---- + ---- + ---- + ----- + ----- - 6 15 13 11 9 7 14 12 10 8 a a a a a a a a a a 4 5 5 5 6 6 6 6 7 7 7 4 z 9 z 15 z 6 z 5 z 17 z 11 z z z z 2 z > ---- - ---- - ----- - ---- - ---- - ----- - ----- + -- + --- + --- + ---- + 6 11 9 7 12 10 8 6 13 11 9 a a a a a a a a a a a 7 8 8 8 9 9 2 z 2 z 5 z 3 z z z > ---- + ---- + ---- + ---- + --- + -- 7 12 10 8 11 9 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 93]], Vassiliev[3][Knot[11, NonAlternating, 93]]}
Out[15]=	{8, 21}
In[16]:=	Kh[Knot[11, NonAlternating, 93]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 2 q t + 3 q t + 2 q t + 3 q t + 3 q t + 5 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 3 q t + 3 q t + 5 q t + 4 q t + 3 q t + 2 q t + 21 7 21 8 23 8 25 9 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n93

K11n92 K11n94