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

K11a175 K11a177

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X14,5,15,6 X16,8,17,7 X18,10,19,9 X2,12,3,11 X22,13,1,14 X20,15,21,16 X8,18,9,17 X6,20,7,19 X10,21,11,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 2 22 20 8 6 10

Alexander Polynomial:

- t-4 + 5t-3 - 13t-2 + 23t-1 - 27 + 23t - 13t2 + 5t3 - t4

Conway Polynomial:

1 - 3z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{111, 2}

Jones Polynomial:

q-3 - 3q-2 + 6q-1 - 10 + 15q - 17q2 + 18q3 - 16q4 + 12q5 - 8q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-8 - q-6 + 2q-4 - q-2 - 1 + 3q2 - 3q4 + 4q6 - q8 + q12 - 3q14 + 3q16 - q18 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 3a-4 + 10a-4z2 + 8a-4z4 + 2a-4z6 - 3a-2 - 13a-2z2 - 14a-2z4 - 6a-2z6 - a-2z8 + 2 + 5z2 + 4z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 5a-7z3 - 12a-7z5 + 7a-7z7 + a-6 - 2a-6z2 + 4a-6z4 - 10a-6z6 + 7a-6z8 - 2a-5z + 12a-5z3 - 14a-5z5 + 2a-5z7 + 4a-5z9 + 3a-4 - 15a-4z2 + 31a-4z4 - 29a-4z6 + 11a-4z8 + a-4z10 - 2a-3z + 6a-3z3 + a-3z5 - 10a-3z7 + 7a-3z9 + 3a-2 - 19a-2z2 + 32a-2z4 - 26a-2z6 + 8a-2z8 + a-2z10 - 2a-1z + 7a-1z3 - 7a-1z5 - 2a-1z7 + 3a-1z9 + 2 - 5z2 + 8z4 - 10z6 + 4z8 - az + 7az3 - 9az5 + 3az7 + 2a2z2 - 3a2z4 + a2z6

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

{0, 1}

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 11176. 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 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 7 3     j = 9               9 5       j = 7             9 7         j = 5           8 9           j = 3         7 9             j = 1       4 9               j = -1     2 6                 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, Alternating, 176]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 176]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[20, 15, 21, 16], > X[8, 18, 9, 17], X[6, 20, 7, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 176]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 176]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 2, 22, 20, 8, 6, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 176]][t]
Out[6]=	-4 5 13 23 2 3 4 -27 - t + -- - -- + -- + 23 t - 13 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 176]][z]
Out[7]=	4 6 8 1 - 3 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 176]}
In[9]:=	{KnotDet[Knot[11, Alternating, 176]], KnotSignature[Knot[11, Alternating, 176]]}
Out[9]=	{111, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 176]][q]
Out[10]=	-3 3 6 2 3 4 5 6 7 8 -10 + q - -- + - + 15 q - 17 q + 18 q - 16 q + 12 q - 8 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 176]}
In[12]:=	A2Invariant[Knot[11, Alternating, 176]][q]
Out[12]=	-8 -6 2 -2 2 4 6 8 12 14 16 -1 + q - q + -- - q + 3 q - 3 q + 4 q - q + q - 3 q + 3 q - 4 q 18 22 24 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 176]][a, z]
Out[13]=	2 2 2 4 4 4 -6 3 3 2 2 z 10 z 13 z 4 z 8 z 14 z 2 - a + -- - -- + 5 z - ---- + ----- - ----- + 4 z - -- + ---- - ----- + 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 8 6 2 z 6 z z > z + ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 176]][a, z]
Out[14]=	2 2 2 -6 3 3 z 2 z 2 z 2 z 2 z 2 z 15 z 2 + a + -- + -- - -- - --- - --- - --- - a z - 5 z + -- - ---- - ----- - 4 2 7 5 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 3 4 19 z 2 2 z 5 z 12 z 6 z 7 z 3 4 6 z > ----- + 2 a z - -- + ---- + ----- + ---- + ---- + 7 a z + 8 z - ---- + 2 9 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 4 z 31 z 32 z 2 4 z 12 z 14 z z 7 z 5 > ---- + ----- + ----- - 3 a z + -- - ----- - ----- + -- - ---- - 9 a z - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 6 4 z 10 z 29 z 26 z 2 6 7 z 2 z 10 z 2 z > 10 z + ---- - ----- - ----- - ----- + a z + ---- + ---- - ----- - ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 7 z 11 z 8 z 4 z 7 z 3 z z z > 3 a z + 4 z + ---- + ----- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 176]], Vassiliev[3][Knot[11, Alternating, 176]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, Alternating, 176]][q, t]
Out[16]=	3 1 2 1 4 2 6 4 q 3 9 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 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 > 8 q t + 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a176

K11a175 K11a177