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

K11a174 K11a176

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 2 22 6 20 10 8

Alexander Polynomial:

t-4 - 5t-3 + 13t-2 - 21t-1 + 25 - 21t + 13t2 - 5t3 + t4

Conway Polynomial:

1 + 2z2 + 3z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a306, ...}

Determinant and Signature:

{105, 0}

Jones Polynomial:

- q-5 + 3q-4 - 6q-3 + 10q-2 - 14q-1 + 17 - 16q + 15q2 - 11q3 + 7q4 - 4q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - 2q-10 + q-8 + q-6 - 2q-4 + 4q-2 - 2 + 3q2 + q4 + 3q8 - 3q10 - q14 - q16 + q18

HOMFLY-PT Polynomial:

2a-4z2 + a-4z4 - 2a-2 - 9a-2z2 - 8a-2z4 - 2a-2z6 + 5 + 14z2 + 14z4 + 6z6 + z8 - 2a2 - 5a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 6a-5z3 - 11a-5z5 + 4a-5z7 - 4a-4z2 + 13a-4z4 - 17a-4z6 + 6a-4z8 - 2a-3z + 13a-3z3 - 11a-3z5 - 4a-3z7 + 4a-3z9 + 2a-2 - 17a-2z2 + 39a-2z4 - 35a-2z6 + 10a-2z8 + a-2z10 - 3a-1z + 7a-1z3 + 3a-1z5 - 12a-1z7 + 7a-1z9 + 5 - 19z2 + 30z4 - 23z6 + 8z8 + z10 - az + 2az3 - 3az5 + 3az9 + 2a2 - 3a2z2 - 3a2z6 + 4a2z8 + a3z - 5a3z5 + 4a3z7 + 3a4z2 - 6a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{2, 0}

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 11175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   4 1   j = 7                 7 3     j = 5               8 4       j = 3             8 7         j = 1           9 8           j = -1         6 9             j = -3       4 8               j = -5     2 6                 j = -7   1 4                   j = -9   2                     j = -11 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, 175]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 175]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[14, 5, 15, 6], X[16, 7, 17, 8], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[6, 15, 7, 16], > X[20, 18, 21, 17], X[10, 20, 11, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 175]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -9, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 175]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 2, 22, 6, 20, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 175]][t]
Out[6]=	-4 5 13 21 2 3 4 25 + t - -- + -- - -- - 21 t + 13 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 175]][z]
Out[7]=	2 4 6 8 1 + 2 z + 3 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 175], Knot[11, Alternating, 306]}
In[9]:=	{KnotDet[Knot[11, Alternating, 175]], KnotSignature[Knot[11, Alternating, 175]]}
Out[9]=	{105, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 175]][q]
Out[10]=	-5 3 6 10 14 2 3 4 5 6 17 - q + -- - -- + -- - -- - 16 q + 15 q - 11 q + 7 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 175]}
In[12]:=	A2Invariant[Knot[11, Alternating, 175]][q]
Out[12]=	-14 -12 2 -8 -6 2 4 2 4 8 10 14 -2 - q + q - --- + q + q - -- + -- + 3 q + q + 3 q - 3 q - q - 10 4 2 q q q 16 18 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 175]][a, z]
Out[13]=	2 2 4 4 2 2 2 2 z 9 z 2 2 4 z 8 z 2 4 5 - -- - 2 a + 14 z + ---- - ---- - 5 a z + 14 z + -- - ---- - 4 a z + 2 4 2 4 2 a a a a a 6 6 2 z 2 6 8 > 6 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 175]][a, z]
Out[14]=	2 2 2 2 2 z 3 z 3 5 2 4 z 17 z 5 + -- + 2 a - --- - --- - a z + a z + a z - 19 z - ---- - ----- - 2 3 a 4 2 a a a a 3 3 3 4 2 2 4 2 6 z 13 z 7 z 3 5 3 4 2 z > 3 a z + 3 a z + ---- + ----- + ---- + 2 a z - 2 a z + 30 z - ---- + 5 3 a 6 a a a 4 4 5 5 5 13 z 39 z 4 4 11 z 11 z 3 z 5 3 5 5 5 > ----- + ----- - 6 a z - ----- - ----- + ---- - 3 a z - 5 a z + a z - 4 2 5 3 a a a a a 6 6 6 7 7 7 6 z 17 z 35 z 2 6 4 6 4 z 4 z 12 z > 23 z + -- - ----- - ----- - 3 a z + 3 a z + ---- - ---- - ----- + 6 4 2 5 3 a a a a a a 8 8 9 9 10 3 7 8 6 z 10 z 2 8 4 z 7 z 9 10 z > 4 a z + 8 z + ---- + ----- + 4 a z + ---- + ---- + 3 a z + z + --- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 175]], Vassiliev[3][Knot[11, Alternating, 175]]}
Out[15]=	{2, 0}
In[16]:=	Kh[Knot[11, Alternating, 175]][q, t]
Out[16]=	9 1 2 1 4 2 6 4 8 6 - + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 > 8 q t + 8 q t + 7 q t + 8 q t + 4 q t + 7 q t + 3 q t + 9 4 9 5 11 5 13 6 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a175

K11a174 K11a176