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

K11n179 K11n181

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,13,4,12 X5,17,6,16 X14,8,15,7 X9,19,10,18 X11,21,12,20 X13,3,14,2 X22,16,1,15 X17,5,18,4 X19,11,20,10 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

3t-3 - 7t-2 + 11t-1 - 13 + 11t - 7t2 + 3t3

Conway Polynomial:

1 + 10z2 + 11z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a339, ...}

Determinant and Signature:

{55, 6}

Jones Polynomial:

q3 - 2q4 + 5q5 - 7q6 + 9q7 - 9q8 + 9q9 - 7q10 + 4q11 - 2q12

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + 2q14 - q16 + q18 + 2q20 + 4q24 - q26 + q28 - q30 - 2q32 - 2q36

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 3a-15z + 3a-15z3 - a-14z2 + 2a-14z4 + a-14z6 + a-13z + 2a-13z7 + 4a-12z2 - 3a-12z4 + 2a-12z8 + a-11z - 4a-11z5 + 2a-11z7 + a-11z9 + 4a-10 - 15a-10z2 + 16a-10z4 - 13a-10z6 + 5a-10z8 - 3a-9z + 7a-9z3 - 10a-9z5 + 2a-9z7 + a-9z9 + 5a-8 - 16a-8z2 + 17a-8z4 - 11a-8z6 + 3a-8z8 + 4a-7z3 - 6a-7z5 + 2a-7z7 + 4a-6z2 - 4a-6z4 + a-6z6

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

{10, 30}

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 11180. 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                   2 j = 23                 2   j = 21               5 2   j = 19             4 2     j = 17           5 5       j = 15         4 4         j = 13       3 5           j = 11     2 4             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, 180]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 180]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 13, 4, 12], X[5, 17, 6, 16], X[14, 8, 15, 7], > X[9, 19, 10, 18], X[11, 21, 12, 20], X[13, 3, 14, 2], X[22, 16, 1, 15], > X[17, 5, 18, 4], X[19, 11, 20, 10], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 180]]
Out[4]=	GaussCode[1, 7, -2, 9, -3, -1, 4, 11, -5, 10, -6, 2, -7, -4, 8, 3, -9, 5, -10, > 6, -11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 180]]
Out[5]=	DTCode[6, -12, -16, 14, -18, -20, -2, 22, -4, -10, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 180]][t]
Out[6]=	3 7 11 2 3 -13 + -- - -- + -- + 11 t - 7 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 180]][z]
Out[7]=	2 4 6 1 + 10 z + 11 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 339], Knot[11, NonAlternating, 180]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 180]], KnotSignature[Knot[11, NonAlternating, 180]]}
Out[9]=	{55, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 180]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 5 q - 7 q + 9 q - 9 q + 9 q - 7 q + 4 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 180]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 180]][q]
Out[12]=	10 12 14 16 18 20 24 26 28 30 32 36 q - q + 2 q - q + q + 2 q + 4 q - q + q - q - 2 q - 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 180]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -4 5 6 z 12 z 4 z 2 z 9 z 4 z 2 z z --- + -- - ---- + ----- + ---- - ---- + ---- + ---- + ---- + -- 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, 180]][a, z]
Out[14]=	2 2 2 2 2 3 4 5 3 z z z 3 z z 4 z 15 z 16 z 4 z 3 z --- + -- - --- + --- + --- - --- - --- + ---- - ----- - ----- + ---- + ---- + 10 8 15 13 11 9 14 12 10 8 6 15 a a a a a a a a a a a a 3 3 4 4 4 4 4 5 5 5 7 z 4 z 2 z 3 z 16 z 17 z 4 z 4 z 10 z 6 z > ---- + ---- + ---- - ---- + ----- + ----- - ---- - ---- - ----- - ---- + 9 7 14 12 10 8 6 11 9 7 a a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 8 z 13 z 11 z z 2 z 2 z 2 z 2 z 2 z 5 z 3 z > --- - ----- - ----- + -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + 14 10 8 6 13 11 9 7 12 10 8 a a a a a a a a a a a 9 9 z z > --- + -- 11 9 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 180]], Vassiliev[3][Knot[11, NonAlternating, 180]]}
Out[15]=	{10, 30}
In[16]:=	Kh[Knot[11, NonAlternating, 180]][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 + 4 q t + 3 q t + 5 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 4 q t + 4 q t + 5 q t + 5 q t + 4 q t + 2 q t + 21 7 21 8 23 8 25 9 > 5 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n180

K11n179 K11n181