10 105

10_104

10_106

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10 105 Quick Notes

10 105 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_12,4,13,3 X_20,8,1,7 X_16,5,17,6 X_6,15,7,16 X_10,17,11,18 X_18,9,19,10 X_8,14,9,13 X_14,20,15,19 X_2,12,3,11
Gauss code	1, -10, 2, -1, 4, -5, 3, -8, 7, -6, 10, -2, 8, -9, 5, -4, 6, -7, 9, -3
Dowker-Thistlethwaite code	4 12 16 20 18 2 8 6 10 14
Conway Notation	[21:20:20]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-3][-9]
Hyperbolic Volume	15.1817
A-Polynomial	See Data:10 105/A-polynomial

[edit Notes for 10 105'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	-2

[edit Notes for 10 105's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-8 t^2+22 t-29+22 t^{-1} -8 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6-2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 91, 2 }
Jones polynomial	[math]\displaystyle{ q^7-4 q^6+8 q^5-12 q^4+15 q^3-15 q^2+14 q-11+7 q^{-1} -3 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +2 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -3 z^2+a^2+ a^{-2} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +11 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8+3 a z^7+6 z^7 a^{-1} +13 z^7 a^{-3} +10 z^7 a^{-5} +a^2 z^6-19 z^6 a^{-2} -3 z^6 a^{-4} +8 z^6 a^{-6} -7 z^6-8 a z^5-24 z^5 a^{-1} -33 z^5 a^{-3} -13 z^5 a^{-5} +4 z^5 a^{-7} -3 a^2 z^4+2 z^4 a^{-2} -9 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -z^4+7 a z^3+18 z^3 a^{-1} +19 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} +3 a^2 z^2+4 z^2 a^{-2} +5 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}-q^6+3 q^4-2 q^2+2 q^{-2} -3 q^{-4} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} - q^{-20} + q^{-22} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-2 q^{44}+6 q^{42}-10 q^{40}+13 q^{38}-13 q^{36}+4 q^{34}+18 q^{32}-47 q^{30}+82 q^{28}-97 q^{26}+75 q^{24}-8 q^{22}-95 q^{20}+200 q^{18}-257 q^{16}+230 q^{14}-110 q^{12}-81 q^{10}+262 q^8-360 q^6+332 q^4-170 q^2-47+233 q^{-2} -311 q^{-4} +242 q^{-6} -67 q^{-8} -131 q^{-10} +261 q^{-12} -259 q^{-14} +124 q^{-16} +92 q^{-18} -293 q^{-20} +397 q^{-22} -359 q^{-24} +181 q^{-26} +73 q^{-28} -324 q^{-30} +472 q^{-32} -465 q^{-34} +308 q^{-36} -48 q^{-38} -210 q^{-40} +372 q^{-42} -381 q^{-44} +242 q^{-46} -23 q^{-48} -174 q^{-50} +266 q^{-52} -210 q^{-54} +44 q^{-56} +152 q^{-58} -277 q^{-60} +283 q^{-62} -164 q^{-64} -29 q^{-66} +203 q^{-68} -305 q^{-70} +301 q^{-72} -199 q^{-74} +53 q^{-76} +85 q^{-78} -173 q^{-80} +192 q^{-82} -159 q^{-84} +96 q^{-86} -26 q^{-88} -29 q^{-90} +57 q^{-92} -66 q^{-94} +54 q^{-96} -33 q^{-98} +16 q^{-100} + q^{-102} -8 q^{-104} +10 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_105.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^7-2 q^5+4 q^3-4 q+3 q^{-1} - q^{-3} +3 q^{-7} -4 q^{-9} +4 q^{-11} -3 q^{-13} + q^{-15} }[/math]
2	[math]\displaystyle{ q^{22}-2 q^{20}-q^{18}+9 q^{16}-7 q^{14}-14 q^{12}+26 q^{10}+q^8-38 q^6+28 q^4+23 q^2-44+10 q^{-2} +33 q^{-4} -26 q^{-6} -12 q^{-8} +24 q^{-10} +5 q^{-12} -26 q^{-14} +2 q^{-16} +36 q^{-18} -27 q^{-20} -24 q^{-22} +46 q^{-24} -11 q^{-26} -31 q^{-28} +28 q^{-30} +2 q^{-32} -15 q^{-34} +8 q^{-36} + q^{-38} -3 q^{-40} + q^{-42} }[/math]
3	[math]\displaystyle{ q^{45}-2 q^{43}-q^{41}+4 q^{39}+6 q^{37}-10 q^{35}-20 q^{33}+15 q^{31}+48 q^{29}-4 q^{27}-88 q^{25}-38 q^{23}+126 q^{21}+108 q^{19}-128 q^{17}-202 q^{15}+86 q^{13}+289 q^{11}-4 q^9-330 q^7-106 q^5+325 q^3+209 q-278 q^{-1} -279 q^{-3} +199 q^{-5} +314 q^{-7} -110 q^{-9} -315 q^{-11} +27 q^{-13} +295 q^{-15} +53 q^{-17} -249 q^{-19} -135 q^{-21} +190 q^{-23} +212 q^{-25} -111 q^{-27} -283 q^{-29} +9 q^{-31} +328 q^{-33} +103 q^{-35} -326 q^{-37} -213 q^{-39} +284 q^{-41} +279 q^{-43} -197 q^{-45} -296 q^{-47} +100 q^{-49} +262 q^{-51} -24 q^{-53} -192 q^{-55} -15 q^{-57} +115 q^{-59} +27 q^{-61} -61 q^{-63} -20 q^{-65} +30 q^{-67} +6 q^{-69} -10 q^{-71} -3 q^{-73} +5 q^{-75} + q^{-77} -3 q^{-79} + q^{-81} }[/math]

A2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{10}-q^6+3 q^4-2 q^2+2 q^{-2} -3 q^{-4} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} - q^{-20} + q^{-22} }[/math]

2,0

[math]\displaystyle{ q^{28}-2 q^{24}-q^{22}+5 q^{20}+5 q^{18}-7 q^{16}-9 q^{14}+9 q^{12}+10 q^{10}-12 q^8-14 q^6+13 q^4+19 q^2-14-11 q^{-2} +19 q^{-4} +5 q^{-6} -13 q^{-8} -3 q^{-10} +9 q^{-12} -5 q^{-14} -4 q^{-16} +11 q^{-18} -4 q^{-20} -12 q^{-22} +12 q^{-24} +15 q^{-26} -20 q^{-28} -10 q^{-30} +20 q^{-32} +7 q^{-34} -17 q^{-36} -9 q^{-38} +16 q^{-40} +4 q^{-42} -10 q^{-44} - q^{-46} +5 q^{-48} +2 q^{-50} -3 q^{-52} - q^{-54} + q^{-56} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}-2 q^{18}+2 q^{16}+4 q^{14}-10 q^{12}+8 q^{10}+9 q^8-24 q^6+18 q^4+12 q^2-32+21 q^{-2} +13 q^{-4} -29 q^{-6} +9 q^{-8} +14 q^{-10} -12 q^{-12} -5 q^{-14} +6 q^{-16} +12 q^{-18} -13 q^{-20} -9 q^{-22} +32 q^{-24} -18 q^{-26} -18 q^{-28} +34 q^{-30} -15 q^{-32} -17 q^{-34} +23 q^{-36} -5 q^{-38} -10 q^{-40} +9 q^{-42} -3 q^{-46} + q^{-48} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^9-q^7+3 q^5-3 q^3+2 q-2 q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} + q^{-9} - q^{-11} +2 q^{-13} - q^{-15} +3 q^{-17} -3 q^{-19} +3 q^{-21} -2 q^{-23} - q^{-27} + q^{-29} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ q^{20}-2 q^{18}+6 q^{16}-10 q^{14}+18 q^{12}-26 q^{10}+33 q^8-38 q^6+40 q^4-36 q^2+26-13 q^{-2} -5 q^{-4} +25 q^{-6} -45 q^{-8} +62 q^{-10} -72 q^{-12} +77 q^{-14} -72 q^{-16} +62 q^{-18} -45 q^{-20} +27 q^{-22} -6 q^{-24} -12 q^{-26} +26 q^{-28} -36 q^{-30} +39 q^{-32} -39 q^{-34} +33 q^{-36} -25 q^{-38} +18 q^{-40} -11 q^{-42} +6 q^{-44} -3 q^{-46} + q^{-48} }[/math]

1,0

[math]\displaystyle{ q^{34}-2 q^{30}-2 q^{28}+4 q^{26}+7 q^{24}-2 q^{22}-14 q^{20}-6 q^{18}+19 q^{16}+20 q^{14}-14 q^{12}-33 q^{10}-2 q^8+40 q^6+22 q^4-32 q^2-36+14 q^{-2} +42 q^{-4} +5 q^{-6} -36 q^{-8} -17 q^{-10} +25 q^{-12} +21 q^{-14} -17 q^{-16} -22 q^{-18} +11 q^{-20} +25 q^{-22} -6 q^{-24} -28 q^{-26} +31 q^{-30} +9 q^{-32} -30 q^{-34} -20 q^{-36} +28 q^{-38} +32 q^{-40} -17 q^{-42} -41 q^{-44} +40 q^{-48} +18 q^{-50} -27 q^{-52} -31 q^{-54} +8 q^{-56} +28 q^{-58} +8 q^{-60} -15 q^{-62} -14 q^{-64} +3 q^{-66} +10 q^{-68} +3 q^{-70} -3 q^{-72} -3 q^{-74} + q^{-78} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-2 q^{44}+6 q^{42}-10 q^{40}+13 q^{38}-13 q^{36}+4 q^{34}+18 q^{32}-47 q^{30}+82 q^{28}-97 q^{26}+75 q^{24}-8 q^{22}-95 q^{20}+200 q^{18}-257 q^{16}+230 q^{14}-110 q^{12}-81 q^{10}+262 q^8-360 q^6+332 q^4-170 q^2-47+233 q^{-2} -311 q^{-4} +242 q^{-6} -67 q^{-8} -131 q^{-10} +261 q^{-12} -259 q^{-14} +124 q^{-16} +92 q^{-18} -293 q^{-20} +397 q^{-22} -359 q^{-24} +181 q^{-26} +73 q^{-28} -324 q^{-30} +472 q^{-32} -465 q^{-34} +308 q^{-36} -48 q^{-38} -210 q^{-40} +372 q^{-42} -381 q^{-44} +242 q^{-46} -23 q^{-48} -174 q^{-50} +266 q^{-52} -210 q^{-54} +44 q^{-56} +152 q^{-58} -277 q^{-60} +283 q^{-62} -164 q^{-64} -29 q^{-66} +203 q^{-68} -305 q^{-70} +301 q^{-72} -199 q^{-74} +53 q^{-76} +85 q^{-78} -173 q^{-80} +192 q^{-82} -159 q^{-84} +96 q^{-86} -26 q^{-88} -29 q^{-90} +57 q^{-92} -66 q^{-94} +54 q^{-96} -33 q^{-98} +16 q^{-100} + q^{-102} -8 q^{-104} +10 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/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["10 105"];

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

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

Out[4]= [math]\displaystyle{ t^3-8 t^2+22 t-29+22 t^{-1} -8 t^{-2} + t^{-3} }[/math]

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

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

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

Out[7]= { 91, 2 }

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

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

Out[8]= [math]\displaystyle{ q^7-4 q^6+8 q^5-12 q^4+15 q^3-15 q^2+14 q-11+7 q^{-1} -3 q^{-2} + q^{-3} }[/math]

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

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

Out[9]= [math]\displaystyle{ z^6 a^{-2} +2 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -3 z^2+a^2+ a^{-2} -1 }[/math]

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

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

Out[10]= [math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +11 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8+3 a z^7+6 z^7 a^{-1} +13 z^7 a^{-3} +10 z^7 a^{-5} +a^2 z^6-19 z^6 a^{-2} -3 z^6 a^{-4} +8 z^6 a^{-6} -7 z^6-8 a z^5-24 z^5 a^{-1} -33 z^5 a^{-3} -13 z^5 a^{-5} +4 z^5 a^{-7} -3 a^2 z^4+2 z^4 a^{-2} -9 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -z^4+7 a z^3+18 z^3 a^{-1} +19 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} +3 a^2 z^2+4 z^2 a^{-2} +5 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1 }[/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{82}{3} }[/math]

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

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

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

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

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

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

[math]\displaystyle{ -\frac{328}{3} }[/math]

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

[math]\displaystyle{ -\frac{1951}{30} }[/math]

[math]\displaystyle{ \frac{1022}{15} }[/math]

[math]\displaystyle{ -\frac{7022}{45} }[/math]

[math]\displaystyle{ \frac{703}{18} }[/math]

[math]\displaystyle{ -\frac{511}{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]2 is the signature of 10 105. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

-3

11

5

1

4

9

7

3

-4

7

8

5

3

5

7

0

3

7

8

-1

1

5

8

3

-1

2

6

-4

-3

1

5

4

-5

2

-2

-7

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, 105]]
Out[2]=	10
In[3]:=	PD[Knot[10, 105]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 8, 1, 7], X[16, 5, 17, 6], X[6, 15, 7, 16], X[10, 17, 11, 18], X[18, 9, 19, 10], X[8, 14, 9, 13], X[14, 20, 15, 19], X[2, 12, 3, 11]]
In[4]:=	GaussCode[Knot[10, 105]]
Out[4]=	GaussCode[1, -10, 2, -1, 4, -5, 3, -8, 7, -6, 10, -2, 8, -9, 5, -4, 6, -7, 9, -3]
In[5]:=	BR[Knot[10, 105]]
Out[5]=	BR[5, {1, 1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -4}]
In[6]:=	alex = Alexander[Knot[10, 105]][t]
Out[6]=	-3 8 22 2 3 -29 + t - -- + -- + 22 t - 8 t + t 2 t t
In[7]:=	Conway[Knot[10, 105]][z]
Out[7]=	2 4 6 1 - z - 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 105], Knot[11, NonAlternating, 163]}
In[9]:=	{KnotDet[Knot[10, 105]], KnotSignature[Knot[10, 105]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[10, 105]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 -11 + q - -- + - + 14 q - 15 q + 15 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[10, 105]}
In[12]:=	A2Invariant[Knot[10, 105]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 10 12 q - q + -- - -- + 2 q - 3 q + 3 q - 2 q + 2 q + q - 4 2 q q 14 16 18 20 22 2 q + 3 q - 2 q - q + q
In[13]:=	Kauffman[Knot[10, 105]][a, z]
Out[13]=	2 2 2 -2 2 z 3 z 4 z 2 3 z 5 z 4 z -1 - a - a - -- - --- - --- - 2 a z + 5 z + ---- + ---- + ---- + 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 2 2 2 z 6 z 19 z 18 z 3 4 z 8 z 3 a z - ---- + ---- + ----- + ----- + 7 a z - z + -- - ---- - 7 5 3 a 8 6 a a a a a 4 4 5 5 5 5 9 z 2 z 2 4 4 z 13 z 33 z 24 z 5 ---- + ---- - 3 a z + ---- - ----- - ----- - ----- - 8 a z - 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 8 z 3 z 19 z 2 6 10 z 13 z 6 z 7 7 z + ---- - ---- - ----- + a z + ----- + ----- + ---- + 3 a z + 6 4 2 5 3 a a a a a a 8 8 9 9 8 7 z 11 z 2 z 2 z 4 z + ---- + ----- + ---- + ---- 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 105]], Vassiliev[3][Knot[10, 105]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 105]][q, t]
Out[15]=	3 1 2 1 5 2 6 5 q 8 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 8 q t + 7 q t + 7 q t + 8 q t + 5 q t + 7 q t + 3 q t + 11 4 11 5 13 5 15 6 5 q t + q t + 3 q t + q t

10 105

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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