10 115

10_114

10_116

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10 115 Quick Notes

10 115 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_14,6,15,5 X_20,15,1,16 X_16,7,17,8 X_8,19,9,20 X_18,11,19,12 X_10,4,11,3 X_4,10,5,9 X_12,17,13,18 X_2,14,3,13
Gauss code	1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3
Dowker-Thistlethwaite code	6 10 14 16 4 18 2 20 12 8
Conway Notation	[8*20.20]

Three dimensional invariants

Symmetry type	Negative amphicheiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	16.638
A-Polynomial	See Data:10 115/A-polynomial

[edit Notes for 10 115's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	[math]\displaystyle{ 1 }[/math]
Topological 4 genus	[math]\displaystyle{ 1 }[/math]
Concordance genus	[math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant	0

[edit Notes for 10 115's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+9 t^2-26 t+37-26 t^{-1} +9 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+3 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature	{ 109, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-9 q^3+14 q^2-17 q+19-17 q^{-1} +14 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -z^4-a^4 z^2+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a z^9+3 z^9 a^{-1} +8 a^2 z^8+8 z^8 a^{-2} +16 z^8+8 a^3 z^7+13 a z^7+13 z^7 a^{-1} +8 z^7 a^{-3} +4 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -26 z^6+a^5 z^5-13 a^3 z^5-34 a z^5-34 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+a^2 z^4+z^4 a^{-2} -5 z^4 a^{-4} +12 z^4-a^5 z^3+8 a^3 z^3+22 a z^3+22 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +2 a^4 z^2-a^2 z^2-z^2 a^{-2} +2 z^2 a^{-4} -6 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-4 q^{10}+2 q^8-q^6-2 q^4+5 q^2-1+5 q^{-2} -2 q^{-4} - q^{-6} +2 q^{-8} -4 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+16 q^{72}-17 q^{70}+8 q^{68}+17 q^{66}-53 q^{64}+98 q^{62}-130 q^{60}+121 q^{58}-62 q^{56}-61 q^{54}+225 q^{52}-360 q^{50}+410 q^{48}-311 q^{46}+62 q^{44}+258 q^{42}-536 q^{40}+646 q^{38}-522 q^{36}+193 q^{34}+206 q^{32}-514 q^{30}+589 q^{28}-396 q^{26}+28 q^{24}+339 q^{22}-530 q^{20}+436 q^{18}-110 q^{16}-314 q^{14}+652 q^{12}-743 q^{10}+555 q^8-133 q^6-361 q^4+759 q^2-907+759 q^{-2} -361 q^{-4} -133 q^{-6} +555 q^{-8} -743 q^{-10} +652 q^{-12} -314 q^{-14} -110 q^{-16} +436 q^{-18} -530 q^{-20} +339 q^{-22} +28 q^{-24} -396 q^{-26} +589 q^{-28} -514 q^{-30} +206 q^{-32} +193 q^{-34} -522 q^{-36} +646 q^{-38} -536 q^{-40} +258 q^{-42} +62 q^{-44} -311 q^{-46} +410 q^{-48} -360 q^{-50} +225 q^{-52} -61 q^{-54} -62 q^{-56} +121 q^{-58} -130 q^{-60} +98 q^{-62} -53 q^{-64} +17 q^{-66} +8 q^{-68} -17 q^{-70} +16 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_115.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+3 q^9-5 q^7+5 q^5-3 q^3+2 q+2 q^{-1} -3 q^{-3} +5 q^{-5} -5 q^{-7} +3 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-3 q^{30}+q^{28}+11 q^{26}-18 q^{24}-9 q^{22}+44 q^{20}-24 q^{18}-43 q^{16}+65 q^{14}-64 q^{10}+42 q^8+27 q^6-45 q^4-2 q^2+37-2 q^{-2} -45 q^{-4} +27 q^{-6} +42 q^{-8} -64 q^{-10} +65 q^{-14} -43 q^{-16} -24 q^{-18} +44 q^{-20} -9 q^{-22} -18 q^{-24} +11 q^{-26} + q^{-28} -3 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+3 q^{61}-q^{59}-7 q^{57}+2 q^{55}+21 q^{53}+4 q^{51}-58 q^{49}-25 q^{47}+108 q^{45}+93 q^{43}-151 q^{41}-226 q^{39}+156 q^{37}+389 q^{35}-73 q^{33}-540 q^{31}-100 q^{29}+629 q^{27}+303 q^{25}-602 q^{23}-495 q^{21}+480 q^{19}+618 q^{17}-298 q^{15}-650 q^{13}+107 q^{11}+599 q^9+81 q^7-504 q^5-239 q^3+379 q+379 q^{-1} -239 q^{-3} -504 q^{-5} +81 q^{-7} +599 q^{-9} +107 q^{-11} -650 q^{-13} -298 q^{-15} +618 q^{-17} +480 q^{-19} -495 q^{-21} -602 q^{-23} +303 q^{-25} +629 q^{-27} -100 q^{-29} -540 q^{-31} -73 q^{-33} +389 q^{-35} +156 q^{-37} -226 q^{-39} -151 q^{-41} +93 q^{-43} +108 q^{-45} -25 q^{-47} -58 q^{-49} +4 q^{-51} +21 q^{-53} +2 q^{-55} -7 q^{-57} - q^{-59} +3 q^{-61} - q^{-63} }[/math]

A2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-4 q^{10}+2 q^8-q^6-2 q^4+5 q^2-1+5 q^{-2} -2 q^{-4} - q^{-6} +2 q^{-8} -4 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }[/math]

2,0

[math]\displaystyle{ q^{42}-q^{40}-3 q^{38}+2 q^{36}+8 q^{34}-17 q^{30}-5 q^{28}+24 q^{26}+4 q^{24}-27 q^{22}-4 q^{20}+33 q^{18}+10 q^{16}-38 q^{14}-q^{12}+26 q^{10}-9 q^8-18 q^6+9 q^4+12 q^2-6+12 q^{-2} +9 q^{-4} -18 q^{-6} -9 q^{-8} +26 q^{-10} - q^{-12} -38 q^{-14} +10 q^{-16} +33 q^{-18} -4 q^{-20} -27 q^{-22} +4 q^{-24} +24 q^{-26} -5 q^{-28} -17 q^{-30} +8 q^{-34} +2 q^{-36} -3 q^{-38} - q^{-40} + q^{-42} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-3 q^{32}+q^{30}+8 q^{28}-15 q^{26}+3 q^{24}+26 q^{22}-35 q^{20}+4 q^{18}+40 q^{16}-49 q^{14}+2 q^{12}+39 q^{10}-35 q^8-5 q^6+26 q^4-q^2-8- q^{-2} +26 q^{-4} -5 q^{-6} -35 q^{-8} +39 q^{-10} +2 q^{-12} -49 q^{-14} +40 q^{-16} +4 q^{-18} -35 q^{-20} +26 q^{-22} +3 q^{-24} -15 q^{-26} +8 q^{-28} + q^{-30} -3 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}+q^{19}+2 q^{15}-4 q^{13}+3 q^{11}-4 q^9+q^7-2 q^5+4 q^3+2 q+2 q^{-1} +4 q^{-3} -2 q^{-5} + q^{-7} -4 q^{-9} +3 q^{-11} -4 q^{-13} +2 q^{-15} + q^{-19} - q^{-21} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+14 q^{28}-25 q^{26}+37 q^{24}-48 q^{22}+57 q^{20}-60 q^{18}+54 q^{16}-41 q^{14}+20 q^{12}+7 q^{10}-37 q^8+67 q^6-90 q^4+109 q^2-114+109 q^{-2} -90 q^{-4} +67 q^{-6} -37 q^{-8} +7 q^{-10} +20 q^{-12} -41 q^{-14} +54 q^{-16} -60 q^{-18} +57 q^{-20} -48 q^{-22} +37 q^{-24} -25 q^{-26} +14 q^{-28} -7 q^{-30} +3 q^{-32} - q^{-34} }[/math]

1,0

[math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+11 q^{46}+q^{44}-20 q^{42}-16 q^{40}+20 q^{38}+37 q^{36}-3 q^{34}-51 q^{32}-25 q^{30}+47 q^{28}+50 q^{26}-25 q^{24}-64 q^{22}-5 q^{20}+59 q^{18}+26 q^{16}-44 q^{14}-37 q^{12}+28 q^{10}+40 q^8-13 q^6-40 q^4+6 q^2+43+6 q^{-2} -40 q^{-4} -13 q^{-6} +40 q^{-8} +28 q^{-10} -37 q^{-12} -44 q^{-14} +26 q^{-16} +59 q^{-18} -5 q^{-20} -64 q^{-22} -25 q^{-24} +50 q^{-26} +47 q^{-28} -25 q^{-30} -51 q^{-32} -3 q^{-34} +37 q^{-36} +20 q^{-38} -16 q^{-40} -20 q^{-42} + q^{-44} +11 q^{-46} +4 q^{-48} -3 q^{-50} -3 q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+16 q^{72}-17 q^{70}+8 q^{68}+17 q^{66}-53 q^{64}+98 q^{62}-130 q^{60}+121 q^{58}-62 q^{56}-61 q^{54}+225 q^{52}-360 q^{50}+410 q^{48}-311 q^{46}+62 q^{44}+258 q^{42}-536 q^{40}+646 q^{38}-522 q^{36}+193 q^{34}+206 q^{32}-514 q^{30}+589 q^{28}-396 q^{26}+28 q^{24}+339 q^{22}-530 q^{20}+436 q^{18}-110 q^{16}-314 q^{14}+652 q^{12}-743 q^{10}+555 q^8-133 q^6-361 q^4+759 q^2-907+759 q^{-2} -361 q^{-4} -133 q^{-6} +555 q^{-8} -743 q^{-10} +652 q^{-12} -314 q^{-14} -110 q^{-16} +436 q^{-18} -530 q^{-20} +339 q^{-22} +28 q^{-24} -396 q^{-26} +589 q^{-28} -514 q^{-30} +206 q^{-32} +193 q^{-34} -522 q^{-36} +646 q^{-38} -536 q^{-40} +258 q^{-42} +62 q^{-44} -311 q^{-46} +410 q^{-48} -360 q^{-50} +225 q^{-52} -61 q^{-54} -62 q^{-56} +121 q^{-58} -130 q^{-60} +98 q^{-62} -53 q^{-64} +17 q^{-66} +8 q^{-68} -17 q^{-70} +16 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 115"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= [math]\displaystyle{ -t^3+9 t^2-26 t+37-26 t^{-1} +9 t^{-2} - t^{-3} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ -z^6+3 z^4+z^2+1 }[/math]

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= [math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 109, 0 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= [math]\displaystyle{ -q^5+4 q^4-9 q^3+14 q^2-17 q+19-17 q^{-1} +14 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }[/math]

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= [math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -z^4-a^4 z^2+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +z^2-a^2- a^{-2} +3 }[/math]

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= [math]\displaystyle{ 3 a z^9+3 z^9 a^{-1} +8 a^2 z^8+8 z^8 a^{-2} +16 z^8+8 a^3 z^7+13 a z^7+13 z^7 a^{-1} +8 z^7 a^{-3} +4 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -26 z^6+a^5 z^5-13 a^3 z^5-34 a z^5-34 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+a^2 z^4+z^4 a^{-2} -5 z^4 a^{-4} +12 z^4-a^5 z^3+8 a^3 z^3+22 a z^3+22 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +2 a^4 z^2-a^2 z^2-z^2 a^{-2} +2 z^2 a^{-4} -6 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]

Vassiliev invariants

V₂ and V₃:

(1, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 4 }[/math]

[math]\displaystyle{ 0 }[/math]

[math]\displaystyle{ 8 }[/math]

[math]\displaystyle{ -\frac{34}{3} }[/math]

[math]\displaystyle{ -\frac{62}{3} }[/math]

[math]\displaystyle{ 0 }[/math]

[math]\displaystyle{ \frac{32}{3} }[/math]

[math]\displaystyle{ 0 }[/math]

[math]\displaystyle{ -\frac{136}{3} }[/math]

[math]\displaystyle{ -\frac{248}{3} }[/math]

[math]\displaystyle{ -\frac{1649}{30} }[/math]

[math]\displaystyle{ \frac{566}{5} }[/math]

[math]\displaystyle{ -\frac{9538}{45} }[/math]

[math]\displaystyle{ \frac{305}{18} }[/math]

[math]\displaystyle{ -\frac{2609}{30} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 10 115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

8

3

5

3

9

6

-3

1

10

8

2

-1

8

10

2

-3

6

9

-3

-5

3

8

5

-7

1

6

-5

-9

3

-11

1

-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 115]]
Out[2]=	10
In[3]:=	PD[Knot[10, 115]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[20, 15, 1, 16], X[16, 7, 17, 8], X[8, 19, 9, 20], X[18, 11, 19, 12], X[10, 4, 11, 3], X[4, 10, 5, 9], X[12, 17, 13, 18], X[2, 14, 3, 13]]
In[4]:=	GaussCode[Knot[10, 115]]
Out[4]=	GaussCode[1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3]
In[5]:=	BR[Knot[10, 115]]
Out[5]=	BR[5, {1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -3, -4}]
In[6]:=	alex = Alexander[Knot[10, 115]][t]
Out[6]=	-3 9 26 2 3 37 - t + -- - -- - 26 t + 9 t - t 2 t t
In[7]:=	Conway[Knot[10, 115]][z]
Out[7]=	2 4 6 1 + z + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 115]}
In[9]:=	{KnotDet[Knot[10, 115]], KnotSignature[Knot[10, 115]]}
Out[9]=	{109, 0}
In[10]:=	J=Jones[Knot[10, 115]][q]
Out[10]=	-5 4 9 14 17 2 3 4 5 19 - q + -- - -- + -- - -- - 17 q + 14 q - 9 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[10, 115]}
In[12]:=	A2Invariant[Knot[10, 115]][q]
Out[12]=	-16 -14 2 4 2 -6 2 5 2 4 6 -1 - q + q + --- - --- + -- - q - -- + -- + 5 q - 2 q - q + 12 10 8 4 2 q q q q q 8 10 12 14 16 2 q - 4 q + 2 q + q - q
In[13]:=	Kauffman[Knot[10, 115]][a, z]
Out[13]=	2 2 -2 2 2 z 5 z 3 2 2 z z 2 2 3 + a + a - --- - --- - 5 a z - 2 a z - 6 z + ---- - -- - a z + 3 a 4 2 a a a 3 3 3 4 2 z 8 z 22 z 3 3 3 5 3 4 2 a z - -- + ---- + ----- + 22 a z + 8 a z - a z + 12 z - 5 3 a a a 4 4 5 5 5 5 z z 2 4 4 4 z 13 z 34 z 5 ---- + -- + a z - 5 a z + -- - ----- - ----- - 34 a z - 4 2 5 3 a a a a a 6 6 7 3 5 5 5 6 4 z 9 z 2 6 4 6 8 z 13 a z + a z - 26 z + ---- - ---- - 9 a z + 4 a z + ---- + 4 2 3 a a a 7 8 9 13 z 7 3 7 8 8 z 2 8 3 z 9 ----- + 13 a z + 8 a z + 16 z + ---- + 8 a z + ---- + 3 a z a 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 115]], Vassiliev[3][Knot[10, 115]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 115]][q, t]
Out[15]=	10 1 3 1 6 3 8 6 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 9 8 3 3 2 5 2 5 3 7 3 ---- + --- + 8 q t + 9 q t + 6 q t + 8 q t + 3 q t + 6 q t + 3 q t q t 7 4 9 4 11 5 q t + 3 q t + q t

10 115

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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