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

K11n178 K11n180

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,10,4,11 X5,20,6,21 X14,8,15,7 X9,2,10,3 X11,19,12,18 X8,14,9,13 X22,16,1,15 X17,13,18,12 X19,4,20,5 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 18t-1 + 25 - 18t + 7t2 - t3

Conway Polynomial:

1 + z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{1071, K11n156, ...}

Determinant and Signature:

{77, 0}

Jones Polynomial:

q-4 - 4q-3 + 8q-2 - 11q-1 + 13 - 13q + 12q2 - 8q3 + 5q4 - 2q5

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

{...}

A2 (sl(3)) Invariant:

q-12 - 2q-10 + 2q-8 + q-6 - 3q-4 + 2q-2 - 3 + 2q2 + q4 + q6 + 4q8 - 2q10 + q12 - 2q16

HOMFLY-PT Polynomial:

- 2a-4 - 2a-4z2 + 5a-2 + 7a-2z2 + 3a-2z4 - 3 - 5z2 - 3z4 - z6 + a2 + a2z2 + a2z4

Kauffman Polynomial:

2a-5z - 6a-5z3 + 3a-5z5 - 2a-4 + 8a-4z2 - 9a-4z4 + 2a-4z6 + a-4z8 + 2a-3z - 7a-3z5 + 3a-3z7 + a-3z9 - 5a-2 + 17a-2z2 - 15a-2z4 - 4a-2z6 + 6a-2z8 - a-1z + 13a-1z3 - 26a-1z5 + 12a-1z7 + a-1z9 - 3 + 12z2 - 16z4 + 2z6 + 5z8 - az + 5az3 - 12az5 + 9az7 - a2 + 3a2z2 - 9a2z4 + 8a2z6 - 2a3z3 + 4a3z5 + a4z4

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

{1, 2}

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=0 is the signature of 11179. 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 = 11                   2 j = 9                 3   j = 7               5 2   j = 5             7 3     j = 3           6 5       j = 1         7 7         j = -1       5 7           j = -3     3 6             j = -5   1 5               j = -7   3                 j = -9 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, 179]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 179]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 10, 4, 11], X[5, 20, 6, 21], X[14, 8, 15, 7], > X[9, 2, 10, 3], X[11, 19, 12, 18], X[8, 14, 9, 13], X[22, 16, 1, 15], > X[17, 13, 18, 12], X[19, 4, 20, 5], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 179]]
Out[4]=	GaussCode[1, 5, -2, 10, -3, -1, 4, -7, -5, 2, -6, 9, 7, -4, 8, -11, -9, 6, -10, > 3, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 179]]
Out[5]=	DTCode[6, -10, -20, 14, -2, -18, 8, 22, -12, -4, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 179]][t]
Out[6]=	-3 7 18 2 3 25 - t + -- - -- - 18 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 179]][z]
Out[7]=	2 4 6 1 + z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 71], Knot[11, NonAlternating, 156], Knot[11, NonAlternating, 179]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 179]], KnotSignature[Knot[11, NonAlternating, 179]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 179]][q]
Out[10]=	-4 4 8 11 2 3 4 5 13 + q - -- + -- - -- - 13 q + 12 q - 8 q + 5 q - 2 q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 179]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 179]][q]
Out[12]=	-12 2 2 -6 3 2 2 4 6 8 10 12 -3 + q - --- + -- + q - -- + -- + 2 q + q + q + 4 q - 2 q + q - 10 8 4 2 q q q q 16 > 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 179]][a, z]
Out[13]=	2 2 4 2 5 2 2 2 z 7 z 2 2 4 3 z 2 4 6 -3 - -- + -- + a - 5 z - ---- + ---- + a z - 3 z + ---- + a z - z 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 179]][a, z]
Out[14]=	2 2 2 5 2 2 z 2 z z 2 8 z 17 z 2 2 -3 - -- - -- - a + --- + --- - - - a z + 12 z + ---- + ----- + 3 a z - 4 2 5 3 a 4 2 a a a a a a 3 3 4 4 6 z 13 z 3 3 3 4 9 z 15 z 2 4 4 4 > ---- + ----- + 5 a z - 2 a z - 16 z - ---- - ----- - 9 a z + a z + 5 a 4 2 a a a 5 5 5 6 6 3 z 7 z 26 z 5 3 5 6 2 z 4 z 2 6 > ---- - ---- - ----- - 12 a z + 4 a z + 2 z + ---- - ---- + 8 a z + 5 3 a 4 2 a a a a 7 7 8 8 9 9 3 z 12 z 7 8 z 6 z z z > ---- + ----- + 9 a z + 5 z + -- + ---- + -- + -- 3 a 4 2 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 179]], Vassiliev[3][Knot[11, NonAlternating, 179]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 179]][q, t]
Out[16]=	7 1 3 1 5 3 6 5 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 5 q t + 7 q t + 3 q t + 5 q t + 2 q t + 3 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n179

K11n178 K11n180