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

K11n176 K11n178

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X16,6,17,5 X18,7,19,8 X20,10,21,9 X11,3,12,2 X8,13,9,14 X4,16,5,15 X22,17,1,18 X12,20,13,19 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-4 + 5t-3 - 11t-2 + 16t-1 - 17 + 16t - 11t2 + 5t3 - t4

Conway Polynomial:

1 + z2 - z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{83, 2}

Jones Polynomial:

q-3 - 4q-2 + 8q-1 - 11 + 14q - 14q2 + 13q3 - 10q4 + 6q5 - 2q6

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

{...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 2q-4 - q-2 + 1 + 2q2 - 3q4 + 4q6 - 3q8 + 3q10 - q14 + 2q16 - 2q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 2a-4 + 5a-4z2 + 4a-4z4 + a-4z6 - a-2 - 5a-2z2 - 8a-2z4 - 5a-2z6 - a-2z8 + 1 + 2z2 + 3z4 + z6

Kauffman Polynomial:

- 2a-7z + 3a-7z3 + a-6 - 5a-6z2 + 6a-6z4 + a-6z6 - 4a-5z + 10a-5z3 - 7a-5z5 + 5a-5z7 + 2a-4 - 12a-4z2 + 23a-4z4 - 17a-4z6 + 7a-4z8 - 4a-3z + 16a-3z3 - 18a-3z5 + 3a-3z7 + 3a-3z9 + a-2 - 10a-2z2 + 27a-2z4 - 34a-2z6 + 13a-2z8 - 3a-1z + 15a-1z3 - 21a-1z5 + 2a-1z7 + 3a-1z9 + 1 - 2z2 + 8z4 - 15z6 + 6z8 - az + 6az3 - 10az5 + 4az7 + a2z2 - 2a2z4 + a2z6

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=2 is the signature of 11177. 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                 4   j = 9               6 2   j = 7             7 4     j = 5           7 6       j = 3         7 7         j = 1       5 8           j = -1     3 6             j = -3   1 5               j = -5   3                 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, 177]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 177]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[16, 6, 17, 5], X[18, 7, 19, 8], > X[20, 10, 21, 9], X[11, 3, 12, 2], X[8, 13, 9, 14], X[4, 16, 5, 15], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 177]]
Out[4]=	GaussCode[1, 6, -2, -8, 3, -1, 4, -7, 5, 2, -6, -10, 7, -11, 8, -3, 9, -4, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 177]]
Out[5]=	DTCode[6, -10, 16, 18, 20, -2, 8, 4, 22, 12, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 177]][t]
Out[6]=	-4 5 11 16 2 3 4 -17 - t + -- - -- + -- + 16 t - 11 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 177]][z]
Out[7]=	2 4 6 8 1 + z - z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 177]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 177]], KnotSignature[Knot[11, NonAlternating, 177]]}
Out[9]=	{83, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 177]][q]
Out[10]=	-3 4 8 2 3 4 5 6 -11 + q - -- + - + 14 q - 14 q + 13 q - 10 q + 6 q - 2 q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 177]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 177]][q]
Out[12]=	-8 2 2 -2 2 4 6 8 10 14 16 1 + q - -- + -- - q + 2 q - 3 q + 4 q - 3 q + 3 q - q + 2 q - 6 4 q q 18 20 22 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 177]][a, z]
Out[13]=	2 2 2 4 4 6 -6 2 -2 2 z 5 z 5 z 4 4 z 8 z 6 z 1 - a + -- - a + 2 z - -- + ---- - ---- + 3 z + ---- - ---- + z + -- - 4 6 4 2 4 2 4 a a a a a a a 6 8 5 z z > ---- - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 177]][a, z]
Out[14]=	2 2 -6 2 -2 2 z 4 z 4 z 3 z 2 5 z 12 z 1 + a + -- + a - --- - --- - --- - --- - a z - 2 z - ---- - ----- - 4 7 5 3 a 6 4 a a a a a a 2 3 3 3 3 4 10 z 2 2 3 z 10 z 16 z 15 z 3 4 6 z > ----- + a z + ---- + ----- + ----- + ----- + 6 a z + 8 z + ---- + 2 7 5 3 a 6 a a a a a 4 4 5 5 5 6 23 z 27 z 2 4 7 z 18 z 21 z 5 6 z > ----- + ----- - 2 a z - ---- - ----- - ----- - 10 a z - 15 z + -- - 4 2 5 3 a 6 a a a a a 6 6 7 7 7 8 8 17 z 34 z 2 6 5 z 3 z 2 z 7 8 7 z 13 z > ----- - ----- + a z + ---- + ---- + ---- + 4 a z + 6 z + ---- + ----- + 4 2 5 3 a 4 2 a a a a a a 9 9 3 z 3 z > ---- + ---- 3 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 177]], Vassiliev[3][Knot[11, NonAlternating, 177]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 177]][q, t]
Out[16]=	3 1 3 1 5 3 6 5 q 3 8 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 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 > 7 q t + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 4 q t + 13 5 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n177

K11n176 K11n178