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

K11n74 K11n76

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X9,20,10,21 X11,16,12,17 X13,6,14,7 X15,18,16,19 X17,12,18,13 X19,22,20,1 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-3 - 7t-2 + 14t-1 - 17 + 14t - 7t2 + 2t3

Conway Polynomial:

1 + 4z2 + 5z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1065, 1077, K11n71, ...}

Determinant and Signature:

{63, -2}

Jones Polynomial:

q-9 - 3q-8 + 5q-7 - 9q-6 + 10q-5 - 10q-4 + 11q-3 - 7q-2 + 5q-1 - 2

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

{K11n71, ...}

A2 (sl(3)) Invariant:

q-28 + q-26 - q-24 - 5q-20 - 3q-18 + 6q-12 + 2q-10 + 4q-8 + q-6 - 2q-4 + q-2 - 2

HOMFLY-PT Polynomial:

- 3a2 - 5a2z2 - 2a2z4 + 11a4 + 18a4z2 + 10a4z4 + 2a4z6 - 9a6 - 10a6z2 - 3a6z4 + 2a8 + a8z2

Kauffman Polynomial:

- 3az + 3az3 + 3a2 - 6a2z2 + 4a2z4 + a2z6 - 11a3z + 19a3z3 - 10a3z5 + 4a3z7 + 11a4 - 20a4z2 + 19a4z4 - 10a4z6 + 4a4z8 - 21a5z + 42a5z3 - 31a5z5 + 8a5z7 + a5z9 + 9a6 - 15a6z2 + 17a6z4 - 18a6z6 + 7a6z8 - 17a7z + 36a7z3 - 31a7z5 + 7a7z7 + a7z9 + 2a8 + a8z2 - a8z4 - 6a8z6 + 3a8z8 - 4a9z + 10a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{4, -5}

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 1175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 1                   2 j = -1                 3   j = -3               5 3   j = -5             6 2     j = -7           4 5       j = -9         6 6         j = -11       3 4           j = -13     2 6             j = -15   1 3               j = -17   2                 j = -19 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, 75]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 75]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[9, 20, 10, 21], X[11, 16, 12, 17], X[13, 6, 14, 7], X[15, 18, 16, 19], > X[17, 12, 18, 13], X[19, 22, 20, 1], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 75]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 9, -7, 3, -8, 6, -9, 8, -10, > 5, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 75]]
Out[5]=	DTCode[4, 8, -14, 2, -20, -16, -6, -18, -12, -22, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 75]][t]
Out[6]=	2 7 14 2 3 -17 + -- - -- + -- + 14 t - 7 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 75]][z]
Out[7]=	2 4 6 1 + 4 z + 5 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 65], Knot[10, 77], Knot[11, NonAlternating, 71], > Knot[11, NonAlternating, 75]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 75]], KnotSignature[Knot[11, NonAlternating, 75]]}
Out[9]=	{63, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 75]][q]
Out[10]=	-9 3 5 9 10 10 11 7 5 -2 + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 71], Knot[11, NonAlternating, 75]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 75]][q]
Out[12]=	-28 -26 -24 5 3 6 2 4 -6 2 -2 -2 + q + q - q - --- - --- + --- + --- + -- + q - -- + q 20 18 12 10 8 4 q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 75]][a, z]
Out[13]=	2 4 6 8 2 2 4 2 6 2 8 2 2 4 -3 a + 11 a - 9 a + 2 a - 5 a z + 18 a z - 10 a z + a z - 2 a z + 4 4 6 4 4 6 > 10 a z - 3 a z + 2 a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 75]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 3 a + 11 a + 9 a + 2 a - 3 a z - 11 a z - 21 a z - 17 a z - 4 a z - 2 2 4 2 6 2 8 2 10 2 3 3 3 > 6 a z - 20 a z - 15 a z + a z + 2 a z + 3 a z + 19 a z + 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 42 a z + 36 a z + 10 a z + 4 a z + 19 a z + 17 a z - a z - 10 4 3 5 5 5 7 5 9 5 2 6 4 6 > 3 a z - 10 a z - 31 a z - 31 a z - 10 a z + a z - 10 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 18 a z - 6 a z + a z + 4 a z + 8 a z + 7 a z + 3 a z + 4 8 6 8 8 8 5 9 7 9 > 4 a z + 7 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 75]], Vassiliev[3][Knot[11, NonAlternating, 75]]}
Out[15]=	{4, -5}
In[16]:=	Kh[Knot[11, NonAlternating, 75]][q, t]
Out[16]=	3 3 1 2 1 3 2 6 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 4 6 6 4 5 6 2 5 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 2 q t 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n75

K11n74 K11n76